Chapter 9 Introduction to t-Tests

S-adenosyl methionine (SAM-e) is a naturally occurring compound in human cells that is thought to have an effect on depression symptoms. Suppose that a researcher is interested in testing SAM-e on patients who are struggling with cancer. He obtains a sample of n = 25 patients and asks each person to take the suggested dosage each day for 4 weeks. At the end of the 4-week period, each individual takes the Beck Depression Inventory (BDI), which is a 21-item, multiple-choice self-report inventory for measuring the severity of depression. The scores from the sample produced a mean of M = 20.8 with a standard deviation of s = 5.18. In the general population of cancer patients, the standardized test is known to have a population mean of μ = 23.2. Because there are no previous studies using SAM-e with this population, the researcher doesn't know how it will affect these patients; therefore, he uses a two-tailed single-sample t test to test the hypothesis. From the following, select the correct null and alternative hypotheses for this study: 2.) Assume that the depression scores among patients taking SAM-e are normally distributed. You will first need to determine the degrees of freedom. There are:

1- H₀: μSAM-eSAM-e = 23.2; H₁: μSAM-eSAM-e ≠ 23.2 2- 24 degrees of freedom

What statements are true about t-tests?

1.) The formula for the t statistic is t = (M - μ) / sM 2.) When you do not know the population standard deviation, you can use the t statistic, assuming all relevant assumptions are satisfied. 3.) The t statistic does not require any knowledge of the population standard deviation. 4.)The t statistic uses the same formula as the z statistic except that the t statistic uses the estimated standard error in the denominator.

With the exception of whether the population standard deviation is known, the assumptions required for hypothesis tests with the t statistic and with the z statistic are essentially the same:

1.The members of the sample must be a random sample from the population. This also ensures that the observations in the sample are independent. 2.Because of the central limit theorem, this distribution will be normal if the population is normal or if the sample size is sufficiently large.

Thus, there are three changes that can increase the value of the t statistic:

1.The obtained difference (M - μ) increases2.The sample standard deviation or variance (s or s²) decreases3.The sample size (n) increases When these three changes go in the opposite direction (M - μ decreases, s or s² increases, or n decreases), the t statistic decreases. Neither the significance level (α) nor a switch in the test (one-tailed or two-tailed) affects the value of the t statistic, but each affects the likelihood of rejecting the null hypothesis.

For a sample of n = 16 that has a sample variance of 400, what is the estimated standard error for the sample?

Estimated standard errorEstimated standard error = = √√ (sample variance/sample size) The estimated standard error is used as an estimate of the population standard error (σMM) when the value of σ is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean (M) and the population mean (μ).

What is the probability of getting the t statistic or something more extreme for the sample size of n = 29? p =0.1403 . What is the probability of getting the t statistic or something more extreme for the sample size of n = 73? p =0.0422 .

For the first t statistic, since the sample had 29 women, the Degrees of Freedom are df = n - 1 = 29 - 1 = 28. Set the slider to 28 and click the radio button with the single orange line. Move the orange vertical line until you can see 1.10 on the axis below the orange line. You can see the probability associated with this t statistic in the bubble with the orange type along the orange line. For the first t statistic, it should read 0.1403. For the second t statistic, since the sample had 73 women, the Degrees of Freedom are df = n - 1 = 73 - 1 = 72. Set the slider to 72 and click the radio button with the single orange line. Move the orange vertical line until you can see 1.75 on the axis below the orange line. You can see the probability associated with this t statistic in the bubble with the orange type along the orange line. For the second t statistic, it should read 0.0422. The t statistic with larger n is less likely. Because a larger n sample produces a more unlikely t, you are more likely to reject the null hypothesis with a larger sample.

The t statistic does not require any knowledge of the population standard deviation

The t statistic does not require any knowledge of the population standard deviation. It uses the same formula as the z statistic except that its denominator contains the estimated standard error of the mean rather than the population standard error. The t statistic could be considered an estimated z-score. It provides a poor estimate of z at small sample sizes but provides an increasingly good estimate of z at larger sample sizes.

You are planning to evaluate the mean of a single continuous variable from a study with a sample of n = 25 using the t statistic. What are the degrees of freedom for the sample?

df= n-1 df= 24 The degrees of freedom determine the number of scores in the sample that are independent and free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are n - 1 degrees of freedom.

With the exception of whether the population standard deviation is known, the necessary assumptions for hypothesis tests with the t statistic and with the z statistic are

essentially the same

the t distribution is more ____ with a smaller n

flatter and spread out The exact shape of a t distribution changes with degrees of freedom (df). When sample sizes (and therefore df values) are small, the t distribution tends to be flatter and more variable than the normal z-score distribution. As the sample size (and the df values) increases, the distribution becomes more bell shaped and more closely resembles a normal (z) distribution.

Now suppose you have a larger sample size n = 73. Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is the new t-statistic is

new estimated standard error is = 1.0297 new t stat = 1.75

The critical region is determined by the critical t scores, which are ±2.064. Everything to the right of the positive critical t score is in the critical region. Everything to the left of the negative critical t score is in the critical region. The t statistic lies in this critical region. Therefore, the null hypothesis is rejected. Therefore, you can conclude that SAM-e has a significant effect on the moods of cancer patients.

reject or fail to reject

The average duration of labor from the first contraction to the birth of the baby in women over 35 who have not previously given birth and who did not use any pharmaceuticals is 16 hours. Suppose you have a sample of 29 women who exercise daily, and who have an average duration of labor of 17.8 hours and a sample variance of 77.4 hours. You want to test the hypothesis that women who exercise daily have a different duration of labor than all women. Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is sM= the t-statistic =

sM = = √s2/n = √77.429/29 = 1.63371.6 t-statistic = M-u/sM = 17.8-16/1.6337 t stat = 1.10

With another study, where you also plan on evaluating a mean using the t statistic, you have a sample of n = 31 that has an SS of 120. What is the variance for the sample?

sample variance = SS/n-1 120(ss)/(df=n-1) 31-1 = 4 Sample varianceSample variance = = s²s² = = SSn - 1SSn - 1 = = SSdfSSdf The sample variance provides an unbiased estimate of the corresponding population variance.

the formula for the t statistic is

t = (M - μ) / sM.