classical approach hypothesis testing calculator

Hypothesis Testing Calculator

Type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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Hypothesis Testing Calculators

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

One Sample Proportion: Left-Tailed

Given : CL To Find : α, -z α/2

The confidence level is in Percent Decimal

Given : α To Find : CL, -z α/2

The level of significance is in Percent Decimal

Given : p, p̂, α Test hypothesis using Classical Approach and P-value Approach

The population proportion is in Percent Decimal

The sample proportion is in Percent Decimal

The sample size is

Given : p, n, x, α Test hypothesis using Classical Approach and P-value Approach

The number of individuals is

Given : p, n, x of a Binomial Distribution; α Test hypothesis using P-value Approach

One Sample Proportion: Right-Tailed

Given : CL To Find : α, z α

Given : α To Find : CL, z α

One Sample Proportion: Two-Tailed

Given : σ, x, n, p Test hypothesis using the Classical Approach and the P-value Approach

The significance level is in Percent Decimal

Given :p, p̂, α Test hypothesis using Classical Approach and P-value Approach

Given : CL, x, n, p Test hypothesis using the Confidence Interval Method

Two Samples Proportion: Left-Tailed

Given : x 1 , n 1 , x 2 , n 2 , α Test hypothesis using Classical Approach and P-value Approach

The number of individuals from Population 1 is

The sample size from Population 1 is

The number of individuals from Population 2 is

The sample size from Population 2 is

Two Samples Proportion: Right-Tailed

Two samples proportion: two-tailed.

Given : σ, x, n, p Test hypothesis using Classical Approach and P-value Approach

Two Samples Proportion: Confidence Interval Estimate

Given : x 1 , n 1 , x 2 , n 2 , α To : Construct a confidence interval for p 1 - p 2

One Sample Mean: Left-Tailed

Given : CL, df To Find : α, critical t

The degrees of freedom is

Given : α df To Find : CL, critical t

Given : CL, n To Find : α, critical t

Given : α, n To Find : CL, critical t

Given : Raw dataset $X$ (from Sample) To calculate: sample size, mean, standard deviation

Please separate each value with a comma. Do not put a comma or a period at the end.

Given : sample size, test statistic To Calculate : degrees of freedom, P-value

The test statistic is

Given : μ x̄ , x̄, s, n, α Test hypothesis using Classical Approach and P-value Approach

The population mean is

The sample mean is

The sample standard deviation is

Given : μ x̄ , x̄, σ, n, α Test hypothesis using Classical Approach and P-value Approach

The population standard deviation is

One Sample Mean: Right-Tailed

One sample mean: two-tailed.

Given : μ x̄ , x̄, s, n, CL Test Hypothesis using the Confidence Interval Method

Two Samples Mean: Left-Tailed

Given : Raw datasets $X_1$ and $X_2$ (Samples) To Calculate : sample sizes, sample means, sample standard deviations, degrees of freedom (several ones: use whatever is applicable) Use for Independent Samples

Please separate each value with a comma. Do not put a comma or a period at the end. First Dataset

Please separate each value with a comma. Do not put a comma or a period at the end. Second Dataset

Given : Raw datasets $X_1$ and $X_2$ (Samples) To Calculate : differences between values; sample size, sample mean, and sample standard deviation of the difference, degrees of freedom Use for Dependent Samples

Two Independent Samples

Given: x̄ 1 , x̄ 2 , s 1 , s 2 , n 1 , n 2 , α Where: σ 1 and σ 2 are unknown, and are assumed to be equal Pooled Sample Variance is used Test hypothesis using Classical Approach and P-value Approach

The mean of the first sample is

The mean of the second sample is

The standard deviation of the first sample is

The standard deviation of the second sample is

The size of the first sample is

The size of the second sample is

Two Independent Samples: Confidence Interval Method

Given: x̄ 1 , x̄ 2 , s 1 , s 2 , n 1 , n 2 , CL Where: σ 1 and σ 2 are unknown, and are not assumed to be equal Pooled Sample Variance is used Construct a confidence interval for μ 1 - μ 2

The level of confidence is in Percent Decimal

Two Samples Mean: Right-Tailed

Two samples mean: two-tailed.

Given : Raw datasets $X_1$ and $X_2$ (Samples) To Calculate : sample sizes, sample means, sample standard deviations, degrees of freedom (several ones: use whatever is applicable)

First Dataset

Second Dataset

Given : x̄ 1 , x̄ 2 , s 1 , s 2 , n 1 , n 2 , α Where: σ 1 and σ 2 are unknown, and are assumed to be equal Pooled Sample Variance is used Test hypothesis using Classical Approach and P-value Approach

Two Independent Samples - Confidence Interval Method

Given: x̄ 1 , x̄ 2 , s 1 , s 2 , n 1 , n 2 , α Where: σ 1 and σ 2 are unknown, and are not assumed to be equal Test hypothesis using Classical Approach and P-value Approach

One Sample Standard Deviation: Left-Tailed

Given : n, s, σ, α Test hypothesis using Classical Approach and P-value Approach

Given: Raw dataset $X$ (from Sample) To Calculate: sample size, sample standard deviation

One Sample Standard Deviation: Right-Tailed

One sample standard deviation: two-tailed.

Given : n, s, σ, CL Test Hypothesis using the Confidence Interval Method

Goodness-of-Fit Test: Sample Data: Right-Tailed

Given : CL, df To Find : α, critical Chi-Square

Given : α, df To Find : CL, critical Chi-Square

Given: OV, EV To Calculate: df, Χ 2 (Show all steps), P-value

Please separate each value with a comma. Do not put a comma or a period at the end. $OV$ =

Please separate each value with a comma. Do not put a comma or a period at the end. $EV$ =

$OV - EV$ =

$(OV - EV)^2$ =

$\dfrac{(OV - EV)^2}{E}$ =

$\Sigma \dfrac{(OV - EV)^2}{E}$ =

Given: $OV$, Probabilities, Total Number of Trials OR Given: Benford's Law To Calculate $EV$, $df$, $\chi^2$ (Show all steps), P-value

Please separate each value with a comma. Do not put a comma or a period at the end. Probabilities (in decimals only) =

The total number of trials =

Given : $OV$, $EV$, α (Use 5% if not specified) Test Hypothesis Using Classical Approach and P-value Approach

Please separate each value with a comma. Do not put a comma or a period at the end. OV =

Please separate each value with a comma. Do not put a comma or a period at the end. EV =

Given: $OV$, Probabilities, Total Number of Trials, α (Use 5% if not specified) OR Given: Benford's Law, α (Use 5% if not specified) Test Hypothesis Using Classical Approach and P-value Approach

Contingency Table Test of Independence: Sample Data: Right-Tailed

Given : Contingency Table Test for Independence using the Critical Value Method and the P-value Method

How many rows (actual numeric rows) does your table have?

How many columns (actual numeric columns) does your table have?

Please fill in the actual data values. Leave extra boxes blank, or put zeros.

Σ R 1 = Σ R 2 = Σ R 3 = Σ R 4 = Σ R 5 = Σ R 6 = Σ R 7 =

Σ C 1 = Σ C 2 = Σ C 3 = Σ C 4 = Σ C 5 = Σ C 6 = Σ C 7 =

Σ R = Σ C = N =

The expected frequencies for each data value is listed below. Please ignore the zeros.

One-Way ANOVA: Three Samples of Equal Sizes

Given : Datasets 1, 2, and 3 Calculate the F test statistic (Show all work)

Dataset 1, $X_1$ =

Dataset 2, $X_2$ =

Dataset 3, $X_3$ =

Overall $n$ =

x̄ 1 =

x̄ 2 =

x̄ 3 =

Overall x̄ =

DF for Numerator =

X 1 - x̄ 1 =

(X 1 - x̄ 1 ) 2 =

Σ(X 1 - x̄ 1 ) 2 =

X 2 - x̄ 2 =

(X 2 - x̄ 2 ) 2 =

Σ(X 2 - x̄ 2 ) 2 =

X 3 - x̄ 3 =

(X 3 - x̄ 3 ) 2 =

Σ(X 3 - x̄ 3 ) 2 =

DF for Denominator =

Given: Datasets 1, 2, and 3 Test Hypothesis Using Classical Approach and P-value Approach

Please separate each value with a comma. Do not put a comma or a period at the end. Dataset 1, X 1 =

Please separate each value with a comma. Do not put a comma or a period at the end. Dataset 2, X 2 =

Please separate each value with a comma. Do not put a comma or a period at the end. Dataset 3, X 3 =

Two Samples Correlation: Left-Tailed

Given : datasets X and Y Test hypothesis using the Critical Value Method and the P-value Method

Please type each $x-value$ on a new line. No commas. No spaces. No period. No extra lines. X =

Please type each $y-value$ on a new line. No commas. No spaces. No period. No extra lines. Y =

Two Samples Correlation: Right-Tailed

Two samples correlation: two-tailed.

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Hypothesis Testing Calculator

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t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

Hypothesis Testing

Hypothesis testing calculator, hypothesis testing is a fundamental statistical method used to make inferences or conclusions about a population based on sample data. it provides a structured framework for determining whether there is enough evidence to support or refute a specific claim or hypothesis. whether conducting scientific research, analyzing business data, or evaluating experimental results, hypothesis testing helps you make informed decisions by assessing the validity of assumptions., how to use hypothesis testing tool.

Hypothesis Testing involves a series of steps to evaluate a hypothesis and determine whether to accept or reject it based on sample data. Here’s a simplified guide to conducting hypothesis testing:

• Formulate Hypotheses: Begin by stating the null hypothesis (H0) and alternative hypotheses (H1). The null hypothesis represents the default assumption or no effect, while the alternative hypothesis represents the effect or difference you are testing for.
• Choose the Significance Level: Select a significance level (α), commonly set at 0.05 or 5%. This represents the probability of rejecting the null hypothesis when it is true (Type I error).
• Collect and Analyze Data: Gather sample data relevant to your hypothesis. Perform the appropriate statistical test (e.g., t-test, chi-square test) to analyze the data and calculate the test statistic.
• Calculate the P-value: Determine the p-value, which indicates the probability of observing the test results or more extreme results if the null hypothesis is true.
• Make a Decision: Compare the p-value to the significance level. If the p-value is less than the significance level, reject the null hypothesis. Otherwise, accept it.

Based on your decision, conclude the hypothesis and discuss the implications of your findings.

For example, a manufacturer claims that their light bulbs have a lifespan of 1,000 hours. You want to test if the actual lifespan differs from this claim.

The tool will send you some data when you input your 1000 hours of lifespan. Below, we have discussed the result scenarios and told you the best steps to take. 

• Null Hypothesis (H0): The average lifespan of the light bulbs is 1,000 hours.
• Alternative Hypothesis (H1): The average lifespan of the light bulbs is not 1,000 hours.

Significance Level

• Significance Level (α): 0.05

Analyze Data

• Sample Data: Lifespan of 10 light bulbs (in hours): 995, 1,010, 1,020, 1,015, 1,005, 990, 1,030, 1,025, 1,040, 1,000
• Perform a t-test to compare the sample mean to the claimed mean (1,000 hours).

Make a Decision

• If the p-value from the t-test is less than 0.05, reject the null hypothesis.
• If the null hypothesis is rejected, it means there is significant evidence to suggest the average lifespan differs from 1,000 hours.

How Hypothesis Testing is Helpful?

Supports evidence-based decisions.

Hypothesis Testing provides a structured approach to making data-based decisions. It helps validate claims and assumptions with statistical evidence, leading to more informed choices.

Ensures Rigorous Analysis

By using hypothesis testing, you apply rigorous statistical methods to analyze data. This reduces the risk of making decisions based on unreliable or misleading results.

Facilitates Research and Development

In research and development, hypothesis testing helps evaluate new theories, products, or interventions. It provides a scientific basis for determining their effectiveness or impact.

Improves Business Strategies

Businesses use hypothesis testing to assess the effectiveness of marketing campaigns, product changes, or operational improvements. It helps in optimizing strategies based on data-driven insights.

Enhances Accuracy and Reliability

Hypothesis testing helps maintain accuracy and reliability in research findings and data analysis. It ensures that conclusions drawn are statistically valid and not due to random chance.

Wrapping Up!

In conclusion, Hypothesis Testing is a crucial statistical method for evaluating hypotheses and making data-driven decisions. It provides a systematic approach to testing assumptions and assessing evidence, supporting accurate conclusions in various fields such as research, business, and development. By applying hypothesis testing, you can enhance the validity and reliability of your findings and make well-informed decisions based on statistical analysis.

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Privacy overview.