1.) In the z-score formula as it is used in a hypothesis test,a. Explain what is measured by M – u in the numerator.The numerator in the formula for computing the z-score is the sample mean minus the hypothesized population mean which is the difference between the sample mean and the hypothesis. b. Explain what is measured by the standard error in the denominator.The standard error in the denominator measures the standard distance that should exist between the sample mean and the population mean.2.) The value of the z-score that is obtained for a hypothesis test is influenced by several factors. Some factors influence the 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 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.Decreases the denominator, increases the value of zb. Increase the population standard deviation.Increases the denominator, decreases the value of zc. Increase the difference between the sample mean and the value of u specified in the null hypothesis.Increases the numerator, increases the value of z3.) What happens to the boundaries for the critical region when the alpha level isFor example, from .05 to .01? Also, what happens to the probability of a Type I error when the alpha level is lowered?When the alpha level is lowered, the critical region boundaries increases, the probability of committing type I error is less than if the alpha was at 0.05 since the values needed to reject the null hypothesis must be larger and that the researcher is willing to be wrong 1 in 100 times in saying that there is an effect when there is none.4.) Briefly explain the advantage of using an alpha level of .01 versus a level of .05.In general, what is the disadvantage of using a smaller alpha level?Using an alpha level of .01 means that the researcher is being more careful and rigorous since there are more chances of accepting the null than rejecting it with the increased critical values, moreover, a low alpha is more conservative and tends to need higher sample mean values in order to reject the null hypothesis. An alpha level of .05 is usually the standard alpha value in most social science research, it means that if 95% of the time it is true, then it must be true, it also increases the chances of committing type I error. However, using a smaller alpha level would mean that there is less chance of finding any effect since one must have a greater computed value in order to reject the null hypothesis.5.) Discuss the errors that can be made in hypothesis testing.a. What is a Type I error? Why might it occur?Type I error means that the null hypothesis is rejected thus saying that there is an effect when in reality there is none. This occurs when the sample used in the data may contain extreme scores by chance which will cause the appearance of a treatment effect although there was none.b. What is a Type II error? How does it happen?Type II error means that the values derived from the data would result to the failure of rejecting the null hypothesis when the alternative hypothesis is true, that is there is indeed an effect. Type II errors occur when the sample is very small or when the alpha level is large.6.) State College is evaluating a new English composition course for freshmen. ARandom sample of n = 25 freshman is obtained and the students are placed in theCourse during their first semester. One year later, a writing sample is obtained for each student and the writing samples are graded using a standardized evaluation technique. The average score for the sample is M=76. For the general population of college students, writing scores form a normal distribution with a mean of u= 70.a. If the writing scores for the population have a standard deviation of o = 20, does the sample provide enough evidence to conclude that the new composition course has a significant effect? Assume a two-tailed test with = .05z critical = 1.65, z = 1.5; the new writing course does not have a significant effect.b. If the population standard deviation is 0= 10, is the sample sufficient to demonstrate a significant effect? Again, assume a two-tailed test with =.05.z critical = 1.65, z = 3.0; the new writing course has a significant effectc. Briefly explain why you reached different conclusion for part (a) and part (b).Hypothesis testing is dependent upon the data, when the population standard deviation was lowered, meaning there were less variation in the scores, the more the sample was able to demonstrate a significant effect.7.) A sample of n= 4 individuals is selected from a normal population distribution with u= 70 and 0= 10. A treatment is administered to the individuals in the sample, and after the treatment, the sample mean is found to be M= 75.a. On the basis of the sample data, can you conclude that the treatment has a significanteffect? Use a two-tailed test with = .05.z critical = 1.65, z = 1.0; the treatment has no significant effect.b. Suppose that the sample consisted of n=25 individuals and produced a mean of M =75. Repeat the hypothesis test at the .05 level of significance.z critical = 1.65, z = 2.5; the treatment has a significant effect.c. Compare the results from part (a) and part (b). How does the sample size influence the outcome of a hypothesis test?The larger the sample size, the more the sample mean is similar to the true population mean, if the sample size is very small, the least likely it will represent the true population scores.
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