One- and Two-Sample Tests

Question 1

What is the one-sample Z test for proportions used for?

Answer 1

It is used to evaluate the significance of a difference between one sample proportion and the Null hypothesis value for the true population proportion (p hat - Po).

Question 2

What does the one-sample Z-test for sample proportions require in order to be administered?

Answer 2

This test requires that the sample size is sufficiently large that the Normal approximation for Binomial variables applies [nPo >_ 10 and n(1 - Po) >_ 10]

Question 3

What does the p-value of the one-sample Z test for proportions represent?

Answer 3

It represents the probability of getting the observed difference (p hat - Po) due only to random sampling variation . If the probability is sufficiently small (<0.05), the sample proportion provides sufficient evidence to conclude that the population proportion is not equal to Po, and to reject the Null hypothesis of “no difference”.

Question 4

What is the two-sample Z test for proportions used for?

Answer 4

It is used to evaluate the significance of a difference between two sample proportions (phat 1 - phat2) from independent samples as evidence to test the Null hypothesis that two population proportions are equal (P1 P2) = 0.

Question 5

What does the two-sample Z-test require in order to be administered?

Answer 5

This test requires that the sample sizes for both samples are sufficiently large that the Normal approximation for Binomial variables applies
[nphat1>_ 5, n(1 - phat1) >_ 5, nphat2>_ 5, n(1 - phat2) >_ 5]

Question 6

What does the p-value of the two-sample Z test for proportions represent?

Answer 6

It represents the probability of getting the observed difference between the two sample proportions (phat1 - phat2) due only to random sampling variation. If the probability is sufficiently small (<0.05), the observed difference between the two sample proportions provides sufficient evidence to conclude that the two population proportions are not equal (P1≠P2)

Question 7

How do you calculate the 100*(1 - alpha)% confidence interval for a single population proportion (using sample proportion data) and how do you interpret it?

Answer 7

phat +_ Zalpha/2 (square root of (phat(1 - phat))/n)

Assuming randomized unbiased sampling, 100(1 - alpha)% of all confidence intervals computed by this method will include the true value for the population proportion. Thus, we can be 100(1 - alpha)% confident that any specific confidence interval will include P (the true population proportion).

Question 8

How do you calculate the 100*(1 - alpha)% confidence interval for the difference between two population proportions (using sample proportion data) and how do you interpret it?

Answer 8

phat1 - phat2 +_ Zalpha/2 (square root of (phat1(1 - phat1))/n + phat2*(1 - phat2))/n)

Assuming randomized unbiased sampling, 100(1 - alpha)% of all confidence intervals computed by this method will include the true difference between two population proportions. Thus, we can be 100(1 - alpha)% confident that any specific confidence interval will include P1 - P2

Question 9

What is the one-sample Z-test for sample means used for?

Answer 9

It is used to evaluate the significance of a difference between a single sample mean and a null hypothesis value for the true population mean (x bar - μo).

Question 10

What does the one-sample Z-test for sample means require in order to be administered?

Answer 10

It requires that the population standard deviation (σ sigma) be known or estimated by a sample standard deviation S computed from a large sample (n >_ 100).

Question 11

What is the one-sample t-test for sample means used for?

Answer 11

It is used to evaluate the significance of a difference between a single sample mean and a null hypothesis value for the true population mean (x bar - μo). It is different than the Z-test bases on its requirements.

Question 12

What does the one-sample t-test for sample means require in order to be administered? What is the standard error of the mean?

Answer 12

It does NOT require the knowledge of a population standard deviation or very large sample sizes (n >_ 100). Rather, the spread of the sampling distribution of the mean is estimated by the standard error of the mean, which is computed using the sample standard deviation (=S/square root of n)

Question 13

Write a statement of how to interpret the p-value for a one-tailed test? Use one sample Z test as an example and bold what changes when specifying this for other statistical tests?

Answer 13

The probability P[Z >_ + Z-test statistic] or P[Z <_ - Z-test statistic] is the p-value for the one-tailed test of significance.

You interpret this p-value as follows: this is the probability of obtaining values for the sample mean >_ or <_ the observed x bar due only to random sampling variation, when the true population mean value is equal to the null hypothesis value μo.

Question 14

Write a statement of how to interpret the p-value for a two-tailed test? Use one sample Z test as an example and bold what changes when specifying this for other statistical tests?

Answer 14

The probability of 2 * P[ Z >_ |Ztest|] is the p-value for the two-tailed test of significance.

In a two-tailed test, there was no prior expectation that the difference between x bar and μo would be in one direction or the other.

You interpret this p-value as follows: This is the probability of getting an absolute difference |x bar - μo| greater than or equal to the observed difference due only to random sampling variation, when the true population mean is equal to the null hypothesis value.

Question 15

What does the standard deviation of the mean (sigma x bar) quantify?

Answer 15

It quantifies the expected random sampling variation in the value of x bar, given the amount of variation in the population sigma (σ) and the sample size n.

Question 16

What happens to the sources of bias when the spread of the sampling distribution of the mean is estimated by the standard error of the mean? And what is the domino effect that comes with it?

Answer 16

They increase. It not only random variation among individuals in the population anymore, but also measurement error. This can be caused by sloppy technique or poor training, which causes the variability in the data values to increase. Increasing variability means increasing spread, which in turn, increases the width of the CIs, which in turn reduces precision and the power of the test of significance.

Question 17

What are degrees of freedom?

Answer 17

Refers to the data values that are free to vary after the value of the sample mean has been determined.

Question 18

Do confidence intervals provide info of the effect size of the difference between the sample statistic and the null hypothesis value?

Answer 18

Yes, they provide a measure of the magnitude of the evidence against the null hypothesis.

Question 19

Describe the t-distribution and how it changes as sample size changes?

Answer 19

The t-distribution is symmetric and bell-shaped, but more of the area under the probability density curve is located in the tailes and less around the center as in a Normal distribution. However, as sample size increases, the random sampling variation of the sample standard deviation decreases; hence, S approaches sigma (the population parameter for standard deviation)

Question 20

What is the paired t-test used for?

Answer 20

It is used to evaluate the significance of the mean difference between paired data values from before-after or matched-pairs experimental designs to test the Null hypothesis (the true mean difference is μo = 0.

Question 21

What is a the sample size for the paired t-test? And how does this test reduce random sampling variation?

Answer 21

It is the number of difference (D) values. It does so by not including the variation among individuals in the calculation of the sample standard deviation. Instead, the variation comes from the before and after / matched pair values (changes between individuals might produce large differences; however, similar changes in each individual will produce small variations in the D values).

Question 22

What happens to the power of a paired-test as a result of using D values?

Answer 22

Overall, reducing the spread increases the power to detect small differences, compared to randomized two sample designs with independent and treatment groups.

Question 23

What is observed in the sampling distribution of the mean difference when conducting a paired t test?

Answer 23

Sampling distribution of mean difference (D bar):
Smaller variations = smaller spread.

Question 24

What is the two sample Z-tests used for? And what are the requirements needed to conducted this test?

Answer 24

It is used to evaluate the significance of a difference between two sample means (x bar1 - x bar2) as evidence to test the Null hypothesis that the corresponding means of the two populations are equal (Ho: μ1 - μ1 = 0). This test is used when the population standard deviation sigma1 and sigma2 are known or estimated with both sample sizes >_ 100.

Question 25

What is the variance of the sum (or difference) of two random variables equal to?

Answer 25

It is equal to the sum of their variances.

Question 26

What is the two sample t-tests used for? And what are the requirements needed to conducted this test?

Answer 26

It is used to evaluate the significance of the difference between two sample means (x bar1 - x bar2) as evidence to test the Null hypothesis that the corresponding means of the two populations are equal (Ho: μ1 - μ1 = 0). This test is used when the population standard deviation sigma1 and sigma2 are unknown and estimated by sample standard deviations (standard errors) S1 and S2, with sample sizes < 100.