The independent t-test, also called the two sample t-test, independent-samples t-test or student's t-test, is an inferential statistical test that determines whether there is a statistically significant difference between the means in two unrelated groups.
Null and alternative hypotheses for the independent t-test
The null hypothesis for the independent t-test is that the population means from the two unrelated groups are equal:
H0: u1 = u2
In most cases, we are looking to see if we can show that we can reject the null hypothesis and accept the alternative hypothesis, which is that the population means are not equal:
HA: u1 ≠ u2
To do this, we need to set a significance level (also called alpha) that allows us to either reject or accept the alternative hypothesis. Most commonly, this value is set at 0.05.
What do you need to run an independent t-test?
In order to run an independent t-test, you need the following:
- One independent, categorical variable that has two levels/groups.
- One continuous dependent variable.
Unrelated groups
Unrelated groups, also called unpaired groups or independent groups, are groups in which the cases (e.g., participants) in each group are different. Often we are investigating differences in individuals, which means that when comparing two groups, an individual in one group cannot also be a member of the other group and vice versa. An example would be gender - an individual would have to be classified as either male or female – not both.
The independent t-test requires that the dependent variable is approximately normally distributed within each group.
Note: Technically, it is the residuals that need to be normally distributed, but for an independent t-test, both will give you the same result.
You can test for this using a number of different tests, but the Shapiro-Wilks test of normality or a graphical method, such as a Q-Q Plot, are very common. You can run these tests using SPSS Statistics, the procedure for which can be found in our Testing for Normality guide. However, the t-test is described as a robust test with respect to the assumption of normality. This means that some deviation away from normality does not have a large influence on Type I error rates. The exception to this is if the ratio of the smallest to largest group size is greater than 1.5 (largest compared to smallest).
What to do when you violate the normality assumption
If you find that either one or both of your group's data is not approximately normally distributed and groups sizes differ greatly, you have two options: (1) transform your data so that the data becomes normally distributed (to do this in SPSS Statistics see our guide on Transforming Data), or (2) run the Mann-Whitney U test which is a non-parametric test that does not require the assumption of normality (to run this test in SPSS Statistics see our guide on the Mann-Whitney U Test).
Assumption of homogeneity of variance
The independent t-test assumes the variances of the two groups you are measuring are equal in the population. If your variances are unequal, this can affect the Type I error rate. The assumption of homogeneity of variance can be tested using Levene's Test of Equality of Variances, which is produced in SPSS Statistics when running the independent t-test procedure. If you have run Levene's Test of Equality of Variances in SPSS Statistics, you will get a result similar to that below:
This test for homogeneity of variance provides an F-statistic and a significance value (p-value). We are primarily concerned with the significance value – if it is greater than 0.05 (i.e., p > .05), our group variances can be treated as equal. However, if p < 0.05, we have unequal variances and we have violated the assumption of homogeneity of variances.
Overcoming a violation of the assumption of homogeneity of variance
If the Levene's Test for Equality of Variances is statistically significant, which indicates that the group variances are unequal in the population, you can correct for this violation by not using the pooled estimate for the error term for the t-statistic, but instead using an adjustment to the degrees of freedom using the Welch-Satterthwaite method. In all reality, you will probably never have heard of these adjustments because SPSS Statistics hides this information and simply labels the two options as "Equal variances assumed" and "Equal variances not assumed" without explicitly stating the underlying tests used. However, you can see the evidence of these tests as below:
From the result of Levene's Test for Equality of Variances, we can reject the null hypothesis that there is no difference in the variances between the groups and accept the alternative hypothesis that there is a statistically significant difference in the variances between groups. The effect of not being able to assume equal variances is evident in the final column of the above figure where we see a reduction in the value of the t-statistic and a large reduction in the degrees of freedom (df). This has the effect of increasing the p-value above the critical significance level of 0.05. In this case, we therefore do not accept the alternative hypothesis and accept that there are no statistically significant differences between means. This would not have been our conclusion had we not tested for homogeneity of variances.
When reporting the result of an independent t-test, you need to include the t-statistic value, the degrees of freedom (df) and the significance value of the test (p-value). The format of the test result is: t(df) = t-statistic, p = significance value. Therefore, for the example above, you could report the result as t(7.001) = 2.233, p = 0.061.
Fully reporting your results
In order to provide enough information for readers to fully understand the results when you have run an independent t-test, you should include the result of normality tests, Levene's Equality of Variances test, the two group means and standard deviations, the actual t-test result and the direction of the difference (if any). In addition, you might also wish to include the difference between the groups along with a 95% confidence interval. For example:
Inspection of Q-Q Plots revealed that cholesterol concentration was normally distributed for both groups and that there was homogeneity of variance as assessed by Levene's Test for Equality of Variances. Therefore, an independent t-test was run on the data with a 95% confidence interval (CI) for the mean difference. It was found that after the two interventions, cholesterol concentrations in the dietary group (6.15 ± 0.52 mmol/L) were significantly higher than the exercise group (5.80 ± 0.38 mmol/L) (t(38) = 2.470, p = 0.018) with a difference of 0.35 (95% CI, 0.06 to 0.64) mmol/L.
To know how to run an independent t-test in SPSS Statistics, see our SPSS Statistics Independent-Samples T-Test guide. Alternatively, you can carry out an independent-samples t-test using Excel, R and RStudio.
Home About Us Contact Us Terms & Conditions Privacy & Cookies © 2018 Lund Research LtdAs the sample size gets larger the z value increases therefore we will more likely to reject the null hypothesis less likely to fail to reject the null hypothesis thus the power of the test increases.
Which of the following would happen if you increased the sample size?
Because we have more data and therefore more information our estimate is more precise. As our sample size increases the confidence in our estimate increases our uncertainty decreases and we have greater precision.
Which of the following describes the effect of increasing sample size? … There is little or no effect on measures of effect size but the likelihood of rejecting the null hypothesis increases.
Which of the following correctly describes the effect that decreasing sample size and decreasing the standard deviation have on the power of a hypothesis test?
Which of the following correctly describes the effect that decreasing sample size and decreasing the standard deviation have on the power of a hypothesis test? A decrease in sample size will decrease the power but a decrease in standard deviation will increase the power.
What is the effect of increasing the difference between sample means in a two sample t test?
For the independent-measures t-statistic what is the effect of increasing the difference between sample means? Increase the likelihood of rejecting the null hypothesis and increase measures of effect size.
What is the effect of an increase in the variance for the sample of difference scores?
Standard Error is the square root of the variance. When the variance increases so does the standard error. Since the standard error occurs in the denominator of the t statistic when the standard error increases the value of the t decreases.
How does the standard deviation influence the outcome of a hypothesis test and measures of effect size? Increasing the sample variance reduces the likelihood of rejecting the null hypothesis and reduces measures of effect size.
Which of the following would definitely increase the likelihood of rejecting the null hypothesis?
Which of the following would definitely increase the likelihood of rejecting the null hypothesis? All of the other options will increase the likelihood of rejecting the null hypothesis. A researcher selects a sample from a population with = 30 and uses the sample to evaluate the effect of a treatment.
How does sample variance influence the estimated standard error?
Larger variance decreases the standard error but increases measures of effect size Larger variance Increases the standard error but decreases measures of effect size. Larger variance increases both the standard error and measures of effect size.
How does sample variance influence the estimated standard error and measures of effect size such as r2 and Cohens D?
Question: How does sample variance influence the estimated standard error and measures of effect size such as r2 and Cohen’s d? a. Larger variance decreases both the standard error and measures of effect size. … Larger variance increases both the standard error and measures of effect size.
Our result indicates that as the sample size increases the variance of the sample mean decreases.
What effect does increasing the sample size have on the sample mean?
Therefore as a sample size increases the sample mean and standard deviation will be closer in value to the population mean μ and standard deviation σ .
Why increasing the sample size decreases the variability?
In general larger samples will have smaller variability. This is because as the sample size increases the chance of observing extreme values decreases and the observed values for the statistic will group more closely around the mean of the sampling distribution.
How does increasing the size of the samples increase the power of an experiment?
Increasing sample size makes the hypothesis test more sensitive – more likely to reject the null hypothesis when it is in fact false. Thus it increases the power of the test.
Variability can dramatically reduce your statistical power during hypothesis testing. Statistical power is the probability that a test will detect a difference (or effect) that actually exists. … Even when you can’t reduce the variability you can plan accordingly in order to assure that your study has adequate power.
What is the effect of increasing sample size on bias?
Increasing the sample size tends to reduce the sampling error that is it makes the sample statistic less variable. However increasing sample size does not affect survey bias. A large sample size cannot correct for the methodological problems (undercoverage nonresponse bias etc.) that produce survey bias.
What happens when sample size decreases?
The population mean of the distribution of sample means is the same as the population mean of the distribution being sampled from. … Thus as the sample size increases the standard deviation of the means decreases and as the sample size decreases the standard deviation of the sample means increases.
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How does sample size affect statistical significance?
Higher sample size allows the researcher to increase the significance level of the findings since the confidence of the result are likely to increase with a higher sample size. This is to be expected because larger the sample size the more accurately it is expected to mirror the behavior of the whole group.
How would changes in sample size affect the margin of error assuming all else remained constant?
Therefore you not only have to present summary data but must also be able to explain the results in a manner that is accurate and easy to understand. How would changes in sample size affect the margin of error assuming all else remained constant? … A larger sample size would cause the interval to narrow.
How does the variability of the scores in the sample influence the measures of effect size?
Increasing the sample variance reduces the likelihood of rejecting the null hypothesis and reduces measures of effect size.
Which of the following factors help to determine sample size?
Three factors are used in the sample size calculation and thus determine the sample size for simple random samples. These factors are: 1) the margin of error 2) the confidence level and 3) the proportion (or percentage) of the sample that will chose a given answer to a survey question.
In a research study what is the difference between a census and a sample? … How does the principle of diminishing returns affect decisions about sample size? Larger populations require proportionately larger samples. What is generally recommended sample size for qualitative studies?
How does effect size affect power?
The statistical power of a significance test depends on: • The sample size (n): when n increases the power increases • The significance level (α): when α increases the power increases • The effect size (explained below): when the effect size increases the power increases.
How does sample size affect type 1 error?
Changing the sample size has no effect on the probability of a Type I error. it. not rejected the null hypothesis it has become common practice also to report a P-value.
How does sample size affect Type 2 error?
As the sample size increases the probability of a Type II error (given a false null hypothesis) decreases but the maximum probability of a Type I error (given a true null hypothesis) remains alpha by definition.
t-statistic
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Since the square root of n is the denominator of that fraction as it gets bigger the fraction will get smaller. However this fraction is in turn a denominator. As a result as that denominator gets smaller the second fraction gets bigger. Thus the t-value will get bigger as n gets bigger.
Why does increased N make the t statistic larger?
Higher n leads to smaller standard error that gives higher t-value. Higher t-value means lower p-value infering that the difference between sample-mean (ˉX) and population-mean (μ) is significant (hence we reject the null hypothesis).
The t-distribution is most useful for small sample sizes when the population standard deviation is not known or both. As the sample size increases the t-distribution becomes more similar to a normal distribution.