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Chapter 8: Q85E (page 452)

Question: Find the following probabilities for the standard normal random variable z:

a.P(z>1.46)b.P(z<-1.56)c.P(.67≤z<2.41)d.P(-1.96≤z-.33)e.P(Z≥0)f.P(-2.33<z<1.50)

Short Answer

Expert verified

Answer

A random variable is a mathematical expression of a statistical study's result.

Step by step solution

01

(a) The data is given below

The calculation is given below:

P(z>1.46)=1-P(z≤1.46)=1-0.9279=0.0721

02

(b) The data is given below

The calculation is given below:

P(z<-1.56)=0.0594

03

(c) The data is given below

The calculation is given below:


P(.67≤z<2.41)=P(.67≤z<2.41)-P(z<.0.67)=0.99920-0.7486=0.2434

04

(d) The data is given below

The calculation is given below:

P(-1.96≤z-.33)=P(-1.96≤Z<0)-P(-0.33≤Z<0)=0.4750-0.1293=0.3457

05

(e) The data is given below

The calculation is given below:

P(Z≥0)=1-P(z<0)=1-0.5=.5

06

(f) The data is given below

The calculation is given below:

P(-2.33<z<1.50)=P(-2.33<z<0)+P(0<Z<-1.50)

=0.4901+0.4332=0.9233

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Most popular questions from this chapter

Question: Impact of race on football card values. Refer to the Electronic Journal of Sociology (2007) study of the Impact of race on the value of professional football players’ “rookie” cards, Exercise 12.72 (p. 756). Recall that the sample consisted of 148 rookie cards of NFL players who were inducted into the Football Hall of Fame (HOF). The researchers modelled the natural logarithm of card price (y) as a function of the following independent variables:

Race:x1=1ifblack,0ifwhiteCardavailability:x2=1ifhigh,0iflowCardvintage:x3=yearcardprintedFinalist:x4=naturallogarithmof numberoftimesplayeron finalHOFballotPosition-QB::x5=1ifquarterback,0ifnotPosition-RB:x7=1ifrunningback,0ifnotPosition-WR:x8=1ifwidereceiver,0ifnotPosition-TEx9=1iftightend,0ifnotPosition-DL:x10=1ifdefensivelineman,0ifnotPosition-LB:x11=1iflinebacker,0ifnotPosition-DB:x12=1ifdefensiveback,0ifnot

[Note: For position, offensive lineman is the base level.]

  1. The model E(y)=β0+β1x1+β2x2+β3x3+β4x4+β5x5+β6x6+β7x7+β8x8+β9x9+β10x10+β11x11+β12x12 was fit to the data with the following results:R2=0.705,Ra2=0.681,F=26.9.Interpret the results, practically. Make an inference about the overall adequacy of the model.
  2. Refer to part a. Statistics for the race variable were reported as follows:β^1=-0.147,sβ^1=-0.145,t=-1.014,p-value=0.312 .Use this information to make an inference about the impact of race on the value of professional football players’ rookie cards.
  3. Refer to part a. Statistics for the card vintage variable were reported as follows:β^3=-0.074,sβ^3=0.007,t=-10.92,p-value=.000.Use this information to make an inference about the impact of card vintage on the value of professional football players’ rookie cards.
  4. Write a first-order model for E(y) as a function of card vintage x3and position x5-x12that allows for the relationship between price and vintage to vary depending on position.

Identify the rejection region for each of the following cases. Assume

v1=7andv2=9

a. Ha:σ12<σ22,α=.05

b. Ha:σ12>σ22,α=.01

c. Ha:σ12≠σ22,α=.1withs12>s22

d. Ha:σ12<σ22,α=.025

Ages of self-employed immigrants. Is self-employment for immigrant workers a faster route to economic advancement in the country? This was one of the questions studied in research published in the International Journal of Manpower (Vol. 32, 2011). One aspect of the study involved comparing the ages of self-employed and wage-earning immigrants. The researcher found that in Sweden, native wage earners tend to be younger than self-employed natives. However, immigrant wage earners tend to be older than self-employed immigrants. This inference was based on the table's summary statistics for male Swedish immigrants.

Self-employed immigrants

Wage-earning immigrants

Sample Size

870

84,875

Mean

44.88

46.79

Source: Based on L. Andersson, "Occupational Choice and Returns to Self-Employment Among Immigrants," International Journal of Manpower, Vol. 32, No. 8, 2011 (Table I).

a. Based on the information given, why is it impossible to provide a measure of reliability for the inference "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden"?

b. What information do you need to measure reliability for the inference, part a?

c. Give a value of the test statistic that would conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants if you are willing to risk a Type I error rate of .01.

d. Assume that s, the standard deviation of the ages is the same for both self-employed and wage-earning immigrants. Give an estimate of s that would lead you to conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants using α=0.01 .

e. Is the true value of s likely to be larger or smaller than the one you calculated in part d?

Drug content assessment. Scientists at GlaxoSmithKlineMedicines Research Center used high-performance liquidchromatography (HPLC) to determine the amountof drug in a tablet produced by the company (Analytical

Chemistry, Dec. 15, 2009). Drug concentrations (measuredas a percentage) for 50 randomly selected tablets are listedin the table below and saved in the accompanying file.

a. Descriptive statistics for the drug concentrations areshown at the top of the XLSTAT printout on the nextpage. Use this information to assess whether the dataare approximately normal.

b. An XLSTAT normal probability plot follows. Use thisinformation to assess whether the data are approximatelynormal.

91.28 92.83 89.35 91.90 82.85 94.83 89.83 89.00 84.62

86.96 88.32 91.17 83.86 89.74 92.24 92.59 84.21 89.36

90.96 92.85 89.39 89.82 89.91 92.16 88.67 89.35 86.51

89.04 91.82 93.02 88.32 88.76 89.26 90.36 87.16 91.74

86.12 92.10 83.33 87.61 88.20 92.78 86.35 93.84 91.20

93.44 86.77 83.77 93.19 81.79

Descriptive statistics(Quantitative data)

Statistic

Content

Nbr.of Observation

50

Minimum

81.79

Maximum

94.83

1st Quartile

87.2725

Median

89.375

3rd Quartile

91.88

Mean

89.2906

Variance(n-1)

10.1343

Standard deviation(n-1)

3.1834

4.111 Personnel dexterity tests. Personnel tests are designed to test a job applicant’s cognitive and/or physical abilities. The Wonderlic IQ test is an example of the former; the Purdue Pegboard speed test involving the arrangement of pegs on a peg board is an example of the latter. A particular

dexterity test is administered nationwide by a private testing service. It is known that for all tests administered last year, the distribution of scores was approximately normal with mean 75 and standard deviation 7.5.

a. A particular employer requires job candidates to score at least 80 on the dexterity test. Approximately what percentage of the test scores during the past year exceeded 80?

b. The testing service reported to a particular employer that one of its job candidate’s scores fell at the 98th percentile of the distribution (i.e., approximately 98% of the scores were lower than the candidate’s, and only 2%were higher). What was the candidate’s score?
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