# Normal Distribution, Hypothesis tests

1. In z-score formula as it is used in a hypothesis test

a. Explain what is measured by M- µ in the numerator.

b. Explain what is measured by the standard error in the denominator.

2. The value of the z-score that is obtained for a hypothesis test in influenced by several factors. Some factors influence that size of the numerator of the z-score and other factors influence the size of the standard error in the denominator. For each of the following, indicate whether the factor influences the numerator or the denominator. For each of the following, indicate whether the factor influences the numerator or denominator of the z-score and determine whether the effect would be to increase the value of Z(farther from zero) or decrease the value of Z(closer to zero). In each case, assume that all other components of the z-score remain constant.

a. Increase the sample size

b. Increase the population standard deviation

c. Increase the difference between the sample mean and the value of µ specified in the null hypothesis

3. What happens to the boundaries for the critical region when the alpha level is lowered-for example, from .05 to .01? Also, what happens to be probability of a Type I error when the alpha level is lowered?

4. Briefly explain the advantage of using as alpha level of .01 versus a level of .05. In general, what is the disadvantage of using a smaller alpha level?

5. Discuss the errors that can be made in hypothesis testing.

a. What is a Type I error? Why might it occur?

b. What is a Type II error? How does it happen?

14. Assume that a treatment really does have an effect and that the treatment effect is being evaluated with a hypothesis test. If all other factors are held constant, how is the outcome of the hypothesis test influenced by sample size? To answer this question, do the following two tests and compare the results. For both tests, a sample is selected from a normal population distribution with a mean of µ=60 and a standard deviation of s=10. After the treatment is administered to the individuals in the sample, the sample mean is found to be M=65. In each case, use a two-tailed test with alpha (a) =.05.

a. For the first test, assume the sample consists of n=4 individuals.

b. Compute Cohen's d for a sample of n=4.

c. For the second test, assume the sample consists of n=25 individuals.

d. Compute Cohen's d for a sample of n=25.

e. Explain how the outcome of the hypothesis test is influenced by sample size. How is Cohen's d influenced by sample size?

19. A psychologist has developed a standardized test for measuring the vocabulary skills of 4 years-old children. The scores on the test form a normal distribution with µ=60 and s=10. A researcher would like to use this test to investigate the hypothesis that children who grow up as single children develop vocabulary skills at a faster rate than children in large families. A sample of n=25 single children is obtained, and the mean test score for this sample is M=63.

a. On the basis of this sample, can the researcher conclude that vocabulary skills for single children are significantly better than those of the general population? Use a one-tailed test at the .05 level of significance.

b. Perform the same test assuming that the researcher had used a sample of n=100 single children and obtained the same sample mean, M=63

c. You should find that the larger sample (part b) produces a different conclusion than the smaller sample (part a). Explain how the sample size influences the outcome of the hypothesis test.

#### Solution Preview

1. In z-score formula as it is used in a hypothesis test

a. Explain what is measured by M- µ in the numerator.

M is the sample mean.

µ is the hypothesized population mean.

b. Explain what is measured by the standard error in the denominator.

Standard error is the standard deviation of the distribution of sample means.

The Central Limit Theorem states that:

If random samples are selected from a population with a mean µ and a standard deviation s, as the sample size n increases, the distribution of sample means approaches a normal distribution with mean µ and standard deviation s/square root of n. Standard error (=s/square root of n) is the name given to standard deviation of this distribution.

2. The value of the z-score that is obtained for a hypothesis test in influenced by several factors. Some factors influence that size of the numerator of the z-score and other factors influence the size of the standard error in the denominator. For each of the following, indicate whether the factor influences the numerator or the denominator. For each of the following, indicate whether the factor influences the numerator or denominator of the z-score and determine whether the effect would be to increase the value of Z(farther from zero) or decrease the value of Z(closer to zero). In each case, assume that all other components of the z-score remain constant.

a. Increase the sample size

Z increases

Standard error is inversely proportional to the square root of sample size.

The more is the sample size the lower is the standard error.

Standard error is in the denominator of z calculation.

Hence, the lower the value of standard error, the higher is the z value.

b. Increase the population standard deviation

Z decreases

Standard error is directly proportional to the population standard deviation.

The more is the population standard deviation , the greater is the standard error.

Standard error is in the denominator of z calculation. Hence, the greater the value of standard error, the lower is the z value.

c. Increase the difference between the sample mean and the value of µ specified in the null hypothesis

Z increases

Z is directly proportional to the difference between the sample mean and the value of µ. Therefore z increases when the difference increases.

3. What happens to the boundaries for the critical region when the alpha level is lowered-for example, from .05 to .01? Also, what happens to be probability of a Type I error when the alpha level is lowered?

The boundaries for the critical region are further away from the hypothesized mean when the value of alpha is lowered. (The rejection regions become smaller)

The probability of a Type I error decreases when the value of alpha is lowered.

4. Briefly explain the advantage of using as alpha level of .01 versus a level of .05. In general, what is the disadvantage of using a smaller ...

#### Solution Summary

Answers questions on Normal Distribution, Hypothesis tests, Type I and Type II errors, critical region, significance level, sample size, Cohen's d etc.