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# Exercises with Z-Scores and Percentiles

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Six steps of hypothesis testing are:

Identify populations, comparison distribution, and assumption; state null and research hypothesis; determine characteristics of the comparisons distribution; determine critical values, or cutoffs; calculate test statistic; and make a decision.

Question 7.18: Calculate the following percentages for a z- score using z table: -0.08
(-0.08 = 93.62 (46.81+ 46.81) (above) 6.38 (3.19 + 3.19) (below), 0.08 (at least as extreme as this z score).

Question 7.20: Rewrite percentages as probabilities, or p levels: a. 5% (0.05) b. 83% (0.83) c. 51% (0.51).

Question 7.22: If the critical values for a hypothesis test occur where 2.5% of the distribution is in each tail, what are the cutoffs for z?

Question 7.28: If cutoffs for a z test are -2.58 and 2.58, determine whether you would reject or fail to reject the null hypothesis in each of the following cases: a. z = -0.94 b. z = 2.12

Question 7.32: z distribution and height. 15 year old girl is 58 inches tall, average height for girls is at this age is 63.80 inches, with a standard deviation of 2.66 inches; a. Calculate her z score, b. What percentage of girls are taller than her? c. What percentage of girls are shorter? d. How much would she have to grow to be perfectly average?

Question 7.34: The z statistic and height. An average class of thirty three 15 yr. old girls with an average height of 62.6 inches. Remember, u = 63.8 inches and σ = 2.66 inches; a. Calculate z score, b. How does this sample of girls compare to the distribution of sample means?

Question 7.39: Directional versus no directional hypotheses; for each example, identify whether the research has expressed a directional or a no directional hypotheses; a. A researcher is interested in studying the relation between the use of antibacterial products and the dryness of people's skin. He thinks these products might alter the moisture in skin compared to other products that are not antibacterial; b. A student wonders if grades in a class are in any way related to where a student sits in the classroom. In particular, do students who sit in the front row get better grades, on average, than the general population of students?
1. Fill in the blank with the best word or words.
a. Values of a test statistic beyond which you reject the null hypothesis are called critical values.
b. Critical region is the area in the tails in which the null can be rejected.
c. If your data differ from what you would expect if chance were the only thing operating, you would call your finding statistically significant.
d. A hypothesis test in which the research hypothesis is directional is a (n) one- tailed test.
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.
f. If your z-statistic exceeds the critical cutoff, you can reject the null hypothesis.

The police department of a major city has found that the average height of their 1,250 officers is 71 inches (in.) with  = 2.3 inches.
a. How many officers are at least 75 inches tall? 513 (1250 - 71/2.3).
b. What percent of officers are between 65 and 72 inches tall? 5.2% and 5.76%
c. The top 10% of the officers in terms of height also make higher salaries than the shorter officers. How tall does an officer have to be to get in that top 10% group? (I.e. what is the height marking off the top 10 percent?) 6 ft. tall.

2. The verbal part of the Graduate Record Exam (GRE) has a  of 500 and  = 100. Use the normal distribution to answer the following questions:
a. If you wanted to select only students at or above the 90th percentile, what verbal GRE score would you use as a cutoff score?
b. What verbal GRE score corresponds to a percentile rank of 15%?
c. What's the percentile rank of a GRE score of 628?

3. The police department of a major city has found that the average height of their 1,250 officers is 71 inches (in.) with  = 2.3 inches.
d. How many officers are at least 75 inches tall? 513 (1250 - 71/2.3).
e. What percent of officers are between 65 and 72 inches tall? 5.2% and 5.76%
f. The top 10% of the officers in terms of height also make higher salaries than the shorter officers. How tall does an officer have to be to get in that top 10% group? (I.e. what is the height marking off the top 10 percent?) 6 ft. tall.

4. The verbal part of the Graduate Record Exam (GRE) has a  of 500 and  = 100. Use the normal distribution to answer the following questions:
a. If you wanted to select only students at or above the 90th percentile, what verbal GRE score would you use as a cutoff score?
b. What verbal GRE score corresponds to a percentile rank of 15%?
c. What's the percentile rank of a GRE score of 628?

https://brainmass.com/statistics/probability/exercises-with-z-scores-and-percentiles-587502

#### Solution Summary

This solutions deals with how you work with z-scores and percentiles in a data set and involves a little work with the Central Limit Theorem.

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## Percentiles and Hypothesis Testing with Z-Tests

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 1-7
End-of-chapter problems:

If applicable, remember to show work in your homework document for partial credit.
1) What are the 6 steps of hypothesis testing?
(State the 6 steps)
A) Identify the population, comparision distribution, and assumption
B) State the null and research hypothesis
C) Determine the characteristics of the comparision distribution
D) Determine critical values, or cutoff
E) Calculate the test statistics
F) Make a decision
2) Using the z table in Appendix B, calculate the following percentages for a z score of -0.45
2-a) % above this z score: 53% Work: P(z>-.08) =.5319
2-b) % below this z score: 47% Work: P(z<-.08)= .4681
2-c) At least as extreme as this z score (on either side):
94% Work: P(z>.08)+P(z<.08)= 2x .4681=.9362
3) Rewrite each of the following percentages as probabilities, or p levels:
3-a) 5% = .05
3-b) 95% = .95
3-c) 43% = .43
4) If the critical values, or cutoffs, for a two-tailed z test are -2.05 and +2.05, determine whether you would reject or fail to reject the null hypothesis in each of the following cases:
4a) z = 2.23 Since 2.23 is greater than 2.05, we have to fail the null hypothesis.
4b) z = -0.97 Since -.95 is less than -2.05, we have to reject the null hypothesis
5) Imagine a class of twenty-five 12-year-old girls with an average height of 62 inches. We know that the population mean and standard deviation for this age group of girls is m=59 inches, s = 1.5 inches. (Note that this is a z statistic problem.)
5a) Calculate the z statistic for this sample (not the z score). 62-1.5/59=1.02
5b) How does this sample mean compare to the distribution of sample means? In other words, how does the height of the girls in the sample compare to the height of girls in th general population? The majority of the girls are above the average height.
6) For the following scenarios, identify whether the researcher has expressed a directional or a nondirectional hypothesis:
6a) Social media has changed the levels of closeness in long-distance relationships.
Nondirectional
6b) A professor wonders whether students who eat a healthy breakfast score better on exams in morning courses than those who do not eat a healthy breakfast.
Directional
7) For the following scenario, state the null and research hypotheses in both words and symbolic notation. Symbolic notation must include the symbols "" and "" and a comparison operator (=, ≠, <, >, ≤, ≥), as described in Nolan and Heinzen (2014). Remember to consider whether the hypothesis is nondirectional or directional.
Scenario: A professor wonders whether students who eat a healthy breakfast score better on exams in morning courses than those who do not eat a healthy breakfast.
Null Hypothesis (H0): Symbolic Notation HO: m1=m2
Null Hypothesis:
Written Statement Score of a student who eats healthy equals those who don't.
Research Hypothesis (H1): Symbolic Notation Answer
Research Hypothesis:

________________________________________

Part I: Questions 8a-8g
Fill in the highlighted blanks with the best word or words.
8-a) Values of a test statistic beyond which you reject the null hypothesis are called _Critical values_.
8-b) The _Rejection or Critical Region_ is the area in the tails in which the null can be rejected.
8-c) The probability used to determine the critical values, or cutoffs, in hypothesis testing is known as a _P_ level, also known as alpha.
8-d) If your data differ from what you would expect if chance were the only thing operating, you would call your finding _not significant_.
8-e) A hypothesis test in which the research hypothesis is directional is a(n) _one tailed_ test.
8-f) 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.
8-g) 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 1,200 officers is 71 inches (in.) with  = 2.6 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 72 inches tall? Based on the z score and the z table, what is the officer's percentile? (Hint: See slide 7 of this week's related presentation)

Answer (percentile): Work/reasoning using z table (required):
10b) What is the height (in inches) that marks the 80th percentile for this group of officers? (Hint: See slides 14-16 of this week's related presentation)

Work (required):

10c) What percent of officers are between 68 and 72 inches tall? (Hint: See slide 12 of this week's related presentation)

________________________________________

Part I: Questions 11a-11c

The verbal part of the Graduate Record Exam (GRE) has a  of 500 and  = 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 583?
What is the percentile rank of this z score? (Hint: See slide 7 of this week's related presentation)
11b) What GRE score corresponds to a percentile rank of 25%? (Hint: See slide 17 of this week's related presentation)
11c) If you wanted to select only students at or above the 82nd percentile, what GRE score would you use as a cutoff score (i.e. what GRE score corresponds to this percentile)? (Hint: See slides 14-16 of this week's related presentation)

________________________________________

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.
Call the new percentile variable "IQ rank."
1a) List the first 5 IQ ranks from your file (rows 1-5).
Row 1: .86, 123
Row 2: .81, 119
Row 3: .50,104
Row 4: .99,145
Row 5: .59,108

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)
104
________________________________________

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 = 40 and s = 5:

1) What is the z-score corresponding to a raw score of 32?
-1.6 Work: 32-40/5=-1.6
2) What is the z-score corresponding to a raw score of 50?
2 Work:50-40/5=2
3) If a person has a z-score of 1.8, what is his/her raw score?
32.2 Work: 25+(1.8)(4)=32.2
4) If a person has a z-score of -.63, what is his/her raw score?
22.48 Work: 25+(-.63)(4)=22.48
________________________________________

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)
Bar graph
6) One scale variable with frequencies (when you want to see the general shape of the distribution).
Histogram
7) One scale IV and one scale DV
Scatterplot/dot graph
8) One nominal variable broken down into percentages
Pie charts and bar graphs

Submit Homework 6 by 11:59 p.m. (ET) on Monday of Module/Week 6. Remember to name file appropriately.

Done!

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