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Where Millionaires Live in America. In a 2018 study, Phoenix Marketing International identified Bridgeport, Connecticut; San Jose, California; Washington, D.C.; and Lexington Park, Maryland as the four U.S. cities with the highest percentage of millionaires ( Kiplinger website, https://www.kiplinger.com/slideshow/investing/T064-S001-where-millionaires-live-in-america-2018/index.html). The following data show the following number of millionaires for samples of individuals from each of the four cities. a. What is the estimate of the percentage of millionaires in each of these cities? b. Using a .05 level of significance, test for the equality of the population proportion of millionaires for these four cities. What is the p -value and what is your conclusion?

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Essentials Of Statistics For Busin...

9th Edition
David R. Anderson + 4 others
Publisher: South-Western College Pub
ISBN: 9780357045435

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Section
BuyFindarrow_forward

Essentials Of Statistics For Busin...

9th Edition
David R. Anderson + 4 others
Publisher: South-Western College Pub
ISBN: 9780357045435
Chapter 12, Problem 27SE
Textbook Problem
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Where Millionaires Live in America. In a 2018 study, Phoenix Marketing International identified Bridgeport, Connecticut; San Jose, California; Washington, D.C.; and Lexington Park, Maryland as the four U.S. cities with the highest percentage of millionaires (Kiplinger website, https://www.kiplinger.com/slideshow/investing/T064-S001-where-millionaires-live-in-america-2018/index.html). The following data show the following number of millionaires for samples of individuals from each of the four cities.

Chapter 12, Problem 27SE, Where Millionaires Live in America. In a 2018 study, Phoenix Marketing International identified

  1. a. What is the estimate of the percentage of millionaires in each of these cities?
  2. b. Using a .05 level of significance, test for the equality of the population proportion of millionaires for these four cities. What is the p-value and what is your conclusion?

a.

To determine

Find the estimate of the percentage of millionaires in each of the given cities.

Answer to Problem 27SE

The estimate of the percentage of millionaires in City B, City S, City W and City L are 11%, 9.1%, 8.77%, and 8.5%, respectively.

Explanation of Solution

Calculation:

The number of millionaires in City B, City S, City W and City L are given.

Let p1 be the sample proportion of Millionaire in City B, p2 be the sample proportion of Millionaire in City S, p3 be the sample proportion of Millionaire in City W and p4 be the sample proportion of Millionaire in City L.

The samples size for City B city is 44+356=400.

The sample proportion of Millionaire in City B is,

p1¯=44400=0.11

Thus, the estimate of the percentage of millionaires in City B is 0.11×100=11%.

The samples size for City S is 35+350=385.

The sample proportion of Millionaire in City S is,

p2¯=35385=0.091

Thus, the estimate of the percentage of millionaires in City S city is 0.091×100=9.1%.

The samples size for City W is 35+364=399.

The sample proportion of Millionaire in City W is,

p3¯=35399=0.0877

Thus, the estimate of the percentage of millionaires in City W is 0.0877×100=8.77%.

The samples size for City L is 34+366=400.

The sample proportion of Millionaire in City K is,

p4¯=34400=0.085

Thus, the estimate of the percentage of millionaires in City L is 0.085×100=8.5%.

Therefore, the estimate of the percentage of millionaires in City B, City S, City W and City L are 11%, 9.1%, 8.77%, and 8.5%, respectively.

b.

To determine

Perform a hypothesis test for the equality of the population proportion of millionaires for these four cities at α=0.05 level significance.

Find the p-value and draw conclusion.

Answer to Problem 27SE

There is no difference among the population proportion of millionaires for these four cities.

The p-value is 0.6128.

Explanation of Solution

Calculation:

State the test hypotheses.

Null hypothesis:

 H0:p1=p2=p3=p4

That is, all sample proportions for four cities are equal.

Alternative hypothesis:

 Ha:not all population proportions are equal

That is, not all sample proportions for four cities are equal.

The row and column total is tabulated below:

MillionairesCity BCity SCity WCity LTotal
Yes44353534148
No3563503643661,436
Total4003853994001,584

The formula for expected frequency is given below:

eij=(Row i Total)(Coloumn j Total)Total Sample Size

The expected frequency for each category is calculated as follows:

MillionairesCity BCity SCity WCity L
Yes(148)(400)1,584=37.37(148)(385)1,584=35.97(148)(399)1,584=37.28(148)(400)1,584=37.37
No(1,436)(400)1,584362.63(1,436)(385)1,584349.03(1,436)(399)1,584361.72(1,436)(400)1,584362.63

The formula for chi-square test statistic is given as,

χ2=i=1rj=1c(fijeij)2eij where (i,j)th element denotes the frequency corresponding to (i,j)th cell and r represent the rth row and c represents the cth column.

Therefore, the value of chi-square test statistic is,

χ2={(4437.37)237.37+(3535.97)235.97+(3537.28)237.28+(3437.37)237.37+(356362.63)2362.63+(350349.03)2349.03+(364361.71)2361.72+(366362.63)2362.63}=43.9637.37+0.9435.97+5.2037.28+11.3637.37+43.96362.63+0.49349.03+5.20361.72+11.36362.63=1.18+0.03+0.14+0.30+0.12+0+0.01+0.031.81

Thus, the chi-square test statistic is 1.81

Degrees of freedom:

The degrees of freedom for k populations is defined as df=k1.

Thus, for k=4 the degrees of freedom is,

df=k1=41=3

Thus, the degree of freedom is 3.

Level of significance:

The given level of significance is α=0.05.

p-value:

Software procedure:

Step -by-step software procedure to obtain p-value using EXCEL software is as follows:

  • Open an EXCEL sheet and select cell A1.
  • In cell A1 enter the formula =CHISQ.DIST.RT(1.81,3).
  • Press Enter.
  • Output using EXCEL software is given below:

Essentials Of Statistics For Business & Economics, Chapter 12, Problem 27SE

From the EXCEL output, the p-value is 0.6128.

Rejection rule:

  • If the p-valueα, then reject the null hypothesis.
  • Otherwise, failed to reject the null hypothesis.

Conclusion:

Here, the p-value is greater than the level of significance.

That is, p-value(=0.6128)>α(=0.05)

Thus, the decision is “fail to reject the null hypothesis”.

Therefore, the data do not provide sufficient evidence to conclude that not all sample proportions for four cities are equal.

Thus, there is no difference among the population proportion of millionaires for these four cities.

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Essentials Of Statistics For Business & Economics
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