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PSYC 354 HOMEWORK 6 Percentiles and Hypothesis Testing with Z-Tests | Solution
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Homework 6

Percentiles and Hypothesis Testing with Z-Tests

When submitting this file, be sure the filename includes your full name, course and section. Example:  HW6_JohnDoe_354B01

Be sure you have reviewed this module/week’s lesson and presentations along with the practice data analysis before proceeding to the homework exercises. Complete all analyses in SPSS, then copy and paste your output and graphs into your homework document file. Answer any written questions (such as the text-based questions or the APA Participants section) in the appropriate place within the same file.

Part I: Concepts

Questions 1–4

These questions are based on the Nolan and Heinzen reading and end-of-chapter questions.

Part I: Questions 1-8

End-of-chapter problems:

·         Answer the following Nolan and Heinzen end-of-chapter questions for Chapter 7:

o   7.8, 7.18, 7.20, 7.22, 7.28 [sections (a) and (b)]

o   7.32 [sections (a)–(d)], 7.34, 7.39

o   7.40 [sections (a) and (b)].

·         If applicable, remember to show work in your homework document for partial credit.

1)  Exercise 7.8 (1 pt)

(State the 6 steps)

A) Set up the null hypothesis, H0 and the alternative hypothesis, Ha

B) Choose the appropriate level of significance, α.

C) Identify the test statistic need to be used, i.e. sampling distribution of the test statistic.

D) Obtain the critical value of the test

E) Calculate the test statistic

F) Compare the observed test statistic with critical value(s) and draw conclusion about the null hypothesis

2)  Exercise 7.18 (1 pt each)

2-a) Above this z score

Work: P (z  >  –0.08) = 0.5319

2-b)  Below this z score

Work: P (z  <  –0.08) = 0.4681

2-c)  At least as extreme as this z score

Work: P (z  ≥  0.08) + P (z ≤ –0.08) = 2 x 0.4681 = 0.9362

3)  Exercise 7.20 (1 pt each)

3-a)  0.05

3-b)  0.83

3-c)  0.51

4)  Exercise 7.28 (a & b) (1 pt each)

4a)  Since –2.58 < z = –0.98 < 2.58, we fail to reject the null hypothesis.

4b)  Since –2.58 < z = 2.12 < 2.58, we fail to reject the null hypothesis.

5)

Exercise 7.32 (a, b, c) (1 pt each)

5a)  -2.18

Work:

5b)  98.54%

Work: P (Z > -2.18) = 0.9854 = 98.54%

5c)  1.46%

Work: P (Z< -2.18) = 0.0146 = 1.46%

6)

Exercise 7.34 (a & b) (note that this is a z statistic problem) (1 pt each)

6a)  -2.59

6b)  The majority of the girls in the sample are below the average height of the distribution sample means.

7)

Exercise 7.39 (1 pt each)

7a)  Nondirectional hypothesis

7b)  Directional hypothesis

7c)  Nondirectional hypothesis

8)

Exercise 7.40a: Symbolic notation must include the symbols “m1” and “m2” and a comparison operator (=,  ,  <,  >, , ), as described in Nolan and Heinzen (2014) (4 pts)

Null Hypothesis (H0): Symbolic Notation

Null Hypothesis:

Written Statement

H0: The product nature and the dryness of the skin are not associated.

Research Hypothesis (H1): Symbolic Notation

Research Hypothesis:

Written Statement

8b)  Exercise 7.40b: Symbolic notation must include the symbols “m1” and “m2” and a comparison operator (=,  ,  <,  >, , ), as described in Nolan and Heinzen (2014) (4 pts)

Null Hypothesis (H0): Symbolic Notation

m1 = m2

Null Hypothesis:

Written Statement

There is no significant difference between the mean grades of the students who sit in the front row and the general population of students.

Research Hypothesis (H1): Symbolic Notation

Research Hypothesis:

Written Statement

Part I: Questions 9a-9f

Fill in the highlighted blanks with the best word or words. (1 pt each)

9-a)

Values of a test statistic beyond which you reject the null hypothesis are called critical values.

9-b)

The Rejection or Critical Region is the area in the tails in which the null can be rejected.

9-c)

If your data differ from what you would expect if chance were the only thing operating, you would call your finding no significant.

9-d)

A hypothesis test in which the research hypothesis is directional is a(n) one-tailed test.

9-e)

A hypothesis test in which the research hypothesis specifies that there will be a difference but does not specify the direction of that difference is a(n) two-tailed test.

9-f)

If your z-statistic exceeds the critical cutoff, you can reject the null hypothesis.

Part I: Questions 10a-10c

The police department of a major city has found that the average height of their 1,250 officers is 71 inches (m = 71 in.) with s = 2.3 inches. Use the normal distribution and the formulas and steps in this week’s presentations to answer the following questions:

Note: Showing work is required for this section. Remember that it helps to transfer the raw mean and SD from the description above to the standardized curve shown here (though you don’t need to show this). This helps compare raw and z scores and check your work.

10a)

What is the z score for an officer who is 73 inches tall? Based on the z score and the z table, what is the officer’s percentile?  (4 pts)

Work (required):

Work/reasoning using z table (required):

10b)

What is the height (in inches) that marks the 80th percentile for this group of officers? (2 pts)

Work (required):

10c)

What percent of officers are between 64 and 71 inches tall? (2 pts)

Work (required):

Part I: Questions 11a-11c

The verbal part of the Graduate Record Exam (GRE) has a m of 500 and s = 100. Use the normal distribution and the formulas and steps in this week’s presentations to answer the following questions:

Note: Showing work is required for this section. Remember that it helps to transfer the raw mean and SD from the description above to the standardized curve shown here (though you don’t need to show this). This helps compare raw and z scores and check your work.

11a)

What is the z score for a GRE score of 523?

What is the percentile rank of this z score? (4 pts)

Work (required):

Work (required):

11b)

What verbal GRE score corresponds to a percentile rank of 12%? (2 pts)

Work (required):

11c)

If you wanted to select only students at or above the 85th percentile, what verbal GRE score would you use as a cutoff score (i.e. what GRE score corresponds to this percentile)? (2 pts)

Work (required):

Part II: SPSS Analysis

For this section, you will be using last module/week’s data set containing IQ scores.

Open the file; it should also contain the standardized IQ variable you created last module/week.

Part II:
Question 1a & 1b

Use last week’s HW file that you created using IQ scores, and the SPSS reading and presentation from this week.

·         Using the z-scored IQ variable, create percentile ranks assuming the scores are normally distributed.

o   Call the new percentile variable “IQ rank.”

1a)

List the first 5 IQ ranks from your file (rows 1–5). (5 pts)

Row 1:

Row 2:

Row 3:

Row 4:

Row 5:

1b)

Which raw IQ score seems to best divide the top 50% from the bottom 50% of scores?

(This score can be found by looking carefully over the values in the IQ rank column) (1 pt)

Part III: SPSS Data Entry and Analysis

There is no Part III material this module/week.

Part IV: Cumulative

Data  provided below for respective questions.

Part IV: (Non-SPSS)

Questions 1-4

For a distribution with M = 30 and s = 6:

1) What is the z-score corresponding to a raw score of 20? (2 pts)

Work:

2) What is the z-score corresponding to a raw score of 36? (2 pts)

Work:

3) If a person has a z-score of 1.2, what is his/her raw score?  (2 pts)

Work:

4) If a person has a z-score of -.73, what is his/her raw score? (2 pts)

Work:

Part IV:  (Non-SPSS)

Question 5-8

For the following types of data, state the graph that would be the best choice to display the data.

Two items have more than one correct answer—for these, either answer is acceptable.

5) A nominal independent variable (IV) and a scale dependent variable (DV) (1 pt)

6)  One scale variable with frequencies (when you want to see the general shape of the distribution). (1 pt)

7) One scale IV and one scale DV (1 pt)

8) One nominal variable broken down into percentages (1 pt)