The Test Statistic for One Population Mean Calculator is a calculator that is used when the variable is numerical and only one population or group is being studied.
Let's say that an economist, Economist William German, believes that students who work and go to college only spend, on average, $15 a day on food. The variable, money, is numerical and the population is all students who work and go to college.
The null hypothesis in this example for this economist is, H0= μ= $15. μ represents the average money in dollars amount that all working college students spend a day on food. The economist is claiming that this average amount is equal to $15.
The alternative hypothesis, Ha, is either μ > 15, μ < 15, or μ ≠15. In this example, we'll say that Ha is equal to $20 a day. So the alternative hypothesis is that working college students actually spend $20 a day on food instead of the $15 that the economist believes they spend.
So going back to this example, we use the formula for the test statistic for one population mean, which is, Z= (x - μ0)/(σ/√n).
So in this example, the mean, x, is 15 ($15 for the average working college students spend a day on food). Let's say we are dealing with a sample size of 100 students. And let's say that the standard deviation is 1.7. Plugging these values into the formula gives us, Z= (x - μ0)/(σ/√n)= (20-15)/(1.7/100)= 29.411764705882.
Test Statistic Comparing Two Means Calculator
This Test Statistic Comparing Two Population Means Calculator calculates the test statistic when comparing two population means.
This test is used when the varaible is numerical (for example, cost of goods, earnings, stock price) and two populations or groups are being compared.
Two separate random samples must be selected, one from each population, in order to calculate the resultant test statistic.
The formula to calculate the test statistic comparing two population means is, Z= (x - y)/√(σx2/n1 + σy2/n2).
In order to calculate the statistic, we must calculate the sample means (x and y) and sample standard deviations (σx and σy) for each sample separately. n1 and n2 represent the two sample sizes. The sample sizes do not have to be equal.
We then proceed by finding the difference between the sample means (x - y).
We then compute √(σx2/n1 + σy2/n2), which is the standard error.
We then divide the difference of the means by this standard error.
Let's go over a practice. Let's say we have a random sample of 40 cars from each company in which we are comparing two companies. These are electric cars and we want to measure how far each car can go once fully charged. Suppose the average miles the tesla can go is 200 miles with a standard deviation of 1.2 and the average miles the nissan leaf can go is 190 miles with a standard deviation of 1.5.
Given this data, x= 200, σx= 1.2, y= 190, σy= 1.5, n1= 40, and n2=40.
We now have all our data to compute the test statistic comparing two population means. Plugging these into our formula gives us, Z= (x - y)/√(σx2/n1 + σy2/n2)= (200-190).√((1.2/40) + (1.5/40)= 38.49.
Test Statistic for a Single Population Proportion Calculator
The Test Statistic for a Single Population Proportion calculates the test statistic when the variable is categorial (for example, gender, workers/unemployed, democrats/republicans) and only one population is being studied.
The test is looking at the proportion of individuals in the population who have a certain characteristic- for example the proportion of females who have kids.
The null hypothesis is H0:p=p0, where p0 is a certain claimed value.
The formula for the test statistic for a single proportion is, Z= (ṗ - p0)/√p0(1-p0)/n
á¹— represents the number of people in the same population who have a particular characteristic of interest (for example, the number of women who are currently pregnant in the population). p0 is the claimed value for the null hypothesis. n is the sample size.
So let's say that there are women who go to the OBGYN each month and of them, a doctor claims that 80% of them are currently pregnant at the time of their visit. The population are women who go to the OBGYN and the women who are pregnant are the proportion of the population. The claim is that 80% (p=0.8) are pregnant when they visit. You suspect that the amount of women that are pregnant is less than this (< 0.8). Suppose for that 75 out of 100 women sampled were pregnant who went to the OBGYN office.
Realize that á¹— is equal to the the sample proportion which means in this case it is equal to,á¹—= 75/100= 0.75
Plugging these values into the formula, Z= (ṗ - p0)/√p0(1-p0)/n= (0.75- 0.8)/√(0.8)(1-0.8)/100= -1.25
Test Statistic for Two Population Proportions Calculator
The Test Statistic for Two Population Proportions Calculator calculates the test statistic when the variable being tested is categoritcal and you're interested in the proportion of individuals with a certain characteristic of 2 different categories such as gender (male/female). Let's say we're interested in studying people with sickle cell anemia; this is the certain proportion of the population that we want study. Unlike a single proportion population, we study both categories of this population instead of just one. With a single proportion population, we would study either the male or female population who have sickle cell anemia. With two population proportions, we study two populations or groups (such as males and females with sickle cell anemia).
In order to conduct this test, two separate random samples need to be selected, one from each population (in our case, males and females with sickle cell anemia).
The formula to calculate the test statistic for two population proportions is, Z= ṗ1 - ṗ2/√ṗ(1-ṗ)(1/n1 + 1/n2)
á¹—1 and á¹—2 are the sample proportions. For each sample, the sample size is n1 and n2 (they don't need to be equal). á¹— is the pooled sampled proportion, which is the total number of individuals from both samples who have the characteristic of interest divided by the total number of individuals from both samples (n1 + n2).
So, as an example, let's say that there are a group of test takers. Out of the test takers, 8/241 who drank a lot of coffee before the test failed the exam. 4/220 who did not drink any coffee before the test failed the exam. Let's calculate the test statistic for the two population proportions.
Based on this example, the data is á¹—1= 8/241= 0.033, á¹—2= 4/206= 0.019, n1= 241, n2= 220, and á¹—= (4+8)/(141+206)= 12/347= 0.03458.
Plugging these values into the formula, Z= ṗ1 - ṗ2/√ṗ(1-ṗ)(1/n1 + 1/n2)= (0.033 - 0.019)/√0.03458(1-0.03458)(1/241 + 1/206)= 0.82
To use any of these calculators, a user simply enters in all of the various fields and the resultant test statistic will be shown below.
If you want to calculate the test statistic based on paired data samples, see our Paired t-test Calculator
If you want to calculate the test statistic based on unpaired data samples, see our Unpaired t-test Calculator
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Instructions : Use this T-Test Calculator for two Independent Means calculator to conduct a t-test for two population means (\(\mu_1\) and \(\mu_2\)), with unknown population standard deviations. This test apply when you have two-independent samples, and the population standard deviations \(\sigma_1\) and \(\sigma_2\) and not known. Please select the null and alternative hypotheses, type the significance level, the sample means, the sample standard deviations, the sample sizes, and the results of the t-test for two independent samples will be displayed for you:
More about the t-test for two means so you can better interpret the output presented above: A t-test for two means with unknown population variances and two independent samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)).
More specifically, a t-test uses sample information to assess how plausible it is for the population means \(\mu_1\) and \(\mu_2\) to be equal. The test has two non-overlapping hypotheses, the null and the alternative hypothesis.
The null hypothesis is a statement about the population means, specifically the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis.
Properties of the two sample t-test
The main properties of a two sample t-test for two population means are:
- Depending on our knowledge about the "no effect" situation, the t-test can be two-tailed, left-tailed or right-tailed
- The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
- The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
- In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis
How do you compute the t-statistic for the t test for two independent samples?
The formula for a t-statistic for two population means (with two independent samples), with unknown population variances shows us how to calculate t-test with mean and standard deviation and it depends on whether the population variances are assumed to be equal or not. If the population variances are assumed to be unequal, then the formula is:
\[t = \frac{\bar X_1 - \bar X_2}{\sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} }}\]On the other hand, if the population variances are assumed to be equal, then the formula is:
\[t = \frac{\bar X_1 - \bar X_2}{\sqrt{ \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}(\frac{1}{n_1}+\frac{1}{n_2}) } }\]Normally, the way of knowing whether the population variances must be assumed to be equal or unequal is by using an F-test for equality of variances.
With the above t-statistic, we can compute the corresponding p-value, which allows us to assess whether or not there is a statistically significant difference between two means.
Why is it called t-test for independent samples?
This is because the samples are not related with each other, in a way that the outcomes from one sample are unrelated from the other sample. If the samples are related (for example, you are comparing the answers of husbands and wives, or identical twins), you should use a t-test for paired samples instead .
What if the population standard deviations are known?
The main purpose of this calculator is for comparing two population mean when sigma is unknown for both populations. In case that the population standard deviations are known, then you should use instead this z-test for two means .