The t distribution for df = 4 is flatter and more spread out than the t distribution for df = 20.

43.Two samples from same population probably will have different t statistics even if they arethe same size and have the same mean.44.A sample of n = 4 scores with SS = 48 has a variance of 16 and an estimated standard errorof 2.45.A sample of n = 16 scores with a sample variance of s2= 64 would have an estimatedstandard error of 4 points.46.If random samples, each with n = 20 scores, are selected from a population, and the z-scoreand t statistic are computed for each sample, the t statistics will be more variable than the z-scores.47.If two samples each have the same mean, the same number of scores, and are selected fromthe same population, then they will also have identical t statistics.48.As the sample size is increased, the distribution of t statistics becomes flatter and morespread out.49.The t distribution for df = 4 is flatter and more spread out than the t distribution for df = 20.50.If two samples, each with n = 20 scores, are selected from the same population and bothhave the same mean (M = 53) and the same variance (s2= 12), then they will also have the samet statistic.Instructor Notes-Chapter 1-page 132

51.If other factors are held constant, as the sample size increases, the estimated standard errordecreases.52.If other factors are held constant, as the sample variance increases, the estimated standarderror also increases.53.In general, the larger the value of the sample variance, the greater the likelihood of rejectingthe null hypothesis.54.If other factors are held constant, the bigger the sample is, the greater the likelihood ofrejecting the null hypothesis.55.For a hypothesis test using a t statistic, the boundaries for the critical region will change ifthe sample size is changed.56.For a two-tailed test with α = .05 and a sample of n = 16, the boundaries for the criticalregion are t = ±2.120.57.For a one-tailed test with α = .05 and a sample of n = 9, the critical value for the t statistic ist = 1.860.58.In a hypothesis test, a large value for the sample variance increases the likelihood that youwill find a significant treatment effect.59.If a hypothesis test using a sample of n = 16 scores produces a t statistic of t = 2.15, then thecorrect decision is to reject the null hypothesis for a two-tailed test withα= .05.60.As sample size increases, the critical region boundaries for a two-tailed test withα= .05will move closer to zero.61.For a two-tailed hypothesis test with α = .05 and a sample of n = 25 scores, the boundariesfor the critical region are t = ±2.060.62.Two samples are selected from a population and a treatment is administered to the samples.If both samples have the same mean and the same variance, you are more likely to find asignificant treatment effect with a sample of n = 100 than with a sample of n = 4.63. A research report states “t(15) = 2.31, p < .05.”For this study, the sample had n = 16 scores.

What is the t-distribution?

The t-distribution describes the standardized distances of sample means to the population mean when the population standard deviation is not known, and the observations come from a normally distributed population.

Is the t-distribution the same as the Student’s t-distribution?

Yes.

What’s the key difference between the t- and z-distributions?

The standard normal or z-distribution assumes that you know the population standard deviation. The t-distribution is based on the sample standard deviation.

t-Distribution vs. normal distribution

The t-distribution is similar to a normal distribution. It has a precise mathematical definition. Instead of diving into complex math, let’s look at the useful properties of the t-distribution and why it is important in analyses.

• Like the normal distribution, the t-distribution has a smooth shape.
• Like the normal distribution, the t-distribution is symmetric. If you think about folding it in half at the mean, each side will be the same.
• Like a standard normal distribution (or z-distribution), the t-distribution has a mean of zero.
• The normal distribution assumes that the population standard deviation is known. The t-distribution does not make this assumption.
• The t-distribution is defined by the degrees of freedom. These are related to the sample size.
• 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.

Consider the following graph comparing three t-distributions with a standard normal distribution:

Figure 1: Three t-distributions and a standard normal (z-) distribution.

All of the distributions have a smooth shape. All are symmetric. All have a mean of zero.

The shape of the t-distribution depends on the degrees of freedom. The curves with more degrees of freedom are taller and have thinner tails. All three t-distributions have “heavier tails” than the z-distribution.

You can see how the curves with more degrees of freedom are more like a z-distribution. Compare the pink curve with one degree of freedom to the green curve for the z-distribution. The t-distribution with one degree of freedom is shorter and has thicker tails than the z-distribution. Then compare the blue curve with 10 degrees of freedom to the green curve for the z-distribution. These two distributions are very similar.

A common rule of thumb is that for a sample size of at least 30, one can use the z-distribution in place of a t-distribution. Figure 2 below shows a t-distribution with 30 degrees of freedom and a z-distribution. The figure uses a dotted-line green curve for z, so that you can see both curves. This similarity is one reason why a z-distribution is used in statistical methods in place of a t-distribution when sample sizes are sufficiently large.

Figure 2: z-distribution and t-distribution with 30 degrees of freedom

Tails for hypotheses tests and the t-distribution

When you perform a t-test, you check if your test statistic is a more extreme value than expected from the t-distribution.

For a two-tailed test, you look at both tails of the distribution. Figure 3 below shows the decision process for a two-tailed test. The curve is a t-distribution with 21 degrees of freedom. The value from the t-distribution with α = 0.05/2 = 0.025 is 2.080. For a two-tailed test, you reject the null hypothesis if the test statistic is larger than the absolute value of the reference value. If the test statistic value is either in the lower tail or in the upper tail, you reject the null hypothesis. If the test statistic is within the two reference lines, then you fail to reject the null hypothesis.

Figure 3: Decision process for a two-tailed test

For a one-tailed test, you look at only one tail of the distribution. For example, Figure 4 below shows the decision process for a one-tailed test. The curve is again a t-distribution with 21 degrees of freedom. For a one-tailed test, the value from the t-distribution with α = 0.05 is 1.721. You reject the null hypothesis if the test statistic is larger than the reference value. If the test statistic is below the reference line, then you fail to reject the null hypothesis.

Figure 4: Decision process for a one-tailed test

How to use a t-table

Most people use software to perform the calculations needed for t-tests. But many statistics books still show t-tables, so understanding how to use a table might be helpful. The steps below describe how to use a typical t-table.

1. Identify if the table is for two-tailed or one-tailed tests. Then, decide if you have a one-tailed or a two-tailed test. The columns for a t-table identify different alpha levels.
   If you have a table for a one-tailed test, you can still use it for a two-tailed test. If you set α = 0.05 for your two-tailed test and have only a one-tailed table, then use the column for α = 0.025.
2. Identify the degrees of freedom for your data. The rows of a t-table correspond to different degrees of freedom. Most tables go up to 30 degrees of freedom and then stop. The tables assume people will use a z-distribution for larger sample sizes.
3. Find the cell in the table at the intersection of your α level and degrees of freedom. This is the t-distribution value. Compare your statistic to the t-distribution value and make the appropriate conclusion.

What is the assumption the assumption made for performing the hypothesis test with T distribution?

When the null hypothesis is true the t statistic will have an average value of?

Which combination of factors would definitely increase the width of a confidence interval?

Why are t statistics more variable than z scores quizlet?