Homework 05 - Answers and Points

# Homework 05 - Answers and Points -  ...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview:   Homework Assignment 5    Psychology 60  Spring 2010        ‐ For homework problems requiring calculation, you must show all of your work,  including intermediate steps, in order to receive credit.        1) The value of  z X in a hypothesis test is influenced by a variety of factors. Assuming that all  other variables are held constant, explain how the value of  z X  is influenced by each of the  following:  a. (1 point) Increasing the difference between the sample mean  X  and the untreated  population mean μ (the mean given H0: True)  The value of  z X will be more extreme (further from zero)   b. (1 point) Increasing the population standard deviation, σ  The value of  z X will be less extreme (closer to zero) because   c. (1 point) Increasing the number of scores in the sample, n  The value of  z X will be more extreme (further from zero)   [note: the above is true as long as there is a non­zero mean difference between  the sample mean  X  and the untreated population mean μ]      2) A researcher is investigating the effectiveness of a new study‐skills training program for  elementary school children. A sample of n = 25 third‐grade children is selected to participate  in the program and each child is given a standardized achievement test at the end of the  year. For the regular population of third‐grade children, scores on the test form a normal  distribution with mean of μ = 150 and a standard deviation of σ = 25. The mean for the  sample is  X  = 158. Assuming a non‐directional (two tailed) hypothesis, with  α =0.05:  a. (1 point) Identify the independent and dependent variables  IV: whether a child participates in the study skills training program   DV: Score on the standardized achievement test  b. (1 point) In a sentence, state the null hypothesis being tested  Null: The study skills program will not affect the score on the  standardized achievement test.    c. (1 point) In symbols, state the null hypothesis and the alternative hypothesis  H 0 : µtreatment = µno treatment H 0 : µtreatment = 150     or        H 1 : µtreatment ≠ µno treatment H 1 : µtreatment ≠ 150   d. (1 point) Identify the critical values of  z X . That is, beyond what values will you reject  the null hypothesis?  zcritical : ±  1.96    e. (2 points) Calculate  z X  for the sample mean  zX = X−µ , σX zX = X−µ σ2 n or z X = X−µ 158 − 150 zX = σ 625 n 25 or z X = 158 − 150   25 25   zX = 8 , 5 z X = 1.6     f. (1 point) What decision should be made about the null hypothesis and the  effectiveness of this study‐skills program?  Fail to reject the Null: There is not enough evidence to conclude that the study  skills program affects the score on the standardized achievement test      3) If the alpha level of a study is changed from  α =0.05 to  α =0.01,  a. (1 point) What happens to the boundaries of the critical region?   (report the actual critical values for each  α =0.05 and  α =0.01)    1.96   2.576    b. (1 point) What happens to the probability of a Type I error?  (report the actual probability of a Type I error for each  α =0.05 and  α =0.01)    P(Type I error) will be reduced from 0.05   0.01    c. (1 point) What happens to the probability of a Type II error given H1 is true?  (report ­in general­ what happens to the probability of a Type II error)    P(Type II error) will increase when alpha is reduced.       4) Imagine San Diego State is evaluating a new English composition course for freshmen. A  random sample of n = 25 freshmen is obtained and the students are placed in the course  during their first semester. One year later, a writing sample is obtained for each student and  the wiring samples are graded using a standardized evaluation technique. The mean score  for the sample is  X =76. For the general population of college students, writing scores form a  normal distribution with a mean of μ = 70.  a. (3 points) If the writing scores for the population have a standard deviation of σ =  20, does the sample provide enough evidence to conclude that the new composition  course has a statistically significant effect? Assume a two‐tailed test with  α =0.05.      zX = X−µ , σX zX = X−µ σ2 n or z X = X−µ 76 − 70 zX = σ 400 n 25 or z X = 76 − 70   20 25   zX = 6 , 4 z X = 1.5                   zcritical = ±1.96 , z X < zcritical , Fail to Reject H 0     b. (3 points) If the writing scores for the population have a standard deviation of σ =  10, does the sample provide enough evidence to conclude that the new composition  course has a statistically significant effect? Assume a two‐tailed test with  α =0.05.    X−µ X−µ X−µ 76 − 70 76 − 70 zX = , zX = or z X = zX =   or z X = 2 σ 10 σX 100 σ 25 n 25 n   6 z X = , z X = 3.0                   zcritical = ±1.96 , z X > zcritical , Reject H 0   2     c. (1 point) Comparing your answers in part a and part b, explain how the magnitude of  the standard deviation influences the outcomes of a hypothesis test.  All else equal, the smaller the population standard deviation the larger  the value of the test statistic.     5) A researcher is evaluating the influence of a treatment using a random sample selected from  a normally distribution population with a mean of μ = 80 and a standard deviation of σ = 20.  The researcher expects a 12‐point treatment effect and plans to use a two‐tailed hypothesis  test with  α =0.05.  a. (1 point) Calculate Cohen’s d for this effect. How large (in words) is this effect using  the general guidelines given by Cohen?    X − µ 12 estimated Cohen's d = = = 0.6 ,      σ 20 This is between a medium and large effect size      b. (4 points) Compute the power of the hypothesis test if the researcher uses a sample  size of n = 16 individuals    σ Xcritical = µnull + zcriticalσ X , Xcritical = µnull + zcritical   n   for 12 point increase, critical value for the upper tail would equal:  20 Xcritical = 80 + 1.96 × , Xcritical = 80 + 1.96 × 5, Xcritical = 89.8   16   z= Xcritical − µalternative 89.8 − 92 -2.2 , z= , z= , z = -0.44   σX 5 5   p(z > ­0.44) = 0.6700.               Power = 0.6700      c. (4 points) Compute the power of the hypothesis test if the researcher uses a sample  size of n = 25 individuals.      σ Xcritical = µnull + zcriticalσ X , Xcritical = µnull + zcritical   n   for 12 point increase, critical value for the upper tail would equal:  20 Xcritical = 80 + 1.96 × , Xcritical = 80 + 1.96 × 4, Xcritical = 87.84   25   X − µalternative 87.84 − 92 -4.16 z = critical , z= , z= , z = -1.04   σX 4 4   p(z > ­1.04) = 0.8508.          Power = 0.8508  ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online