Essential Statistics
Essential Statistics
2nd Edition
ISBN: 9781259570643
Author: Navidi
Publisher: MCG
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Chapter 8.4, Problem 10E

a.

To determine

Compute the sample proportion p^.

a.

Expert Solution
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Answer to Problem 10E

The value of sample proportion p^ is 0.633.

Explanation of Solution

In a simple random sample, of 60 samples, only 38 individuals are in the group of interest.

The sample proportion p^ is:

p^=xn

Where, x be the number of observations, n be the sample size.

Substitute the corresponding values to get the proportion,

p^=3860=0.633

Thus, the sample proportion p^ is 0.633.

b.

To determine

Check whether the assumptions for the hypothesis test are satisfied.

b.

Expert Solution
Check Mark

Answer to Problem 10E

Yes, the assumptions for the hypothesis test are satisfied.

Explanation of Solution

Assumptions for performing a hypothesis test for a population proportion:

  • The samples taken from the population are simple random samples.
  • The population is at least 20 times as large as the sample.
  • The samples in the population are divided into two categories.
  • The values of np^0 and nq^0 should be greater than or equal to 10.

Requirement check:

  • The sample of 60 is simple random samples.
  • The information about the population is not known. The population size is assumed to be more than 20 times as large as the sample.
  • The samples in the population seemed to be categorized into two parts. That is, individuals under category of interest and individuals not under the category of interest.
  • Verify the condition: np^010 and nq^010

  Substitute n as 60 and p^0 as 0.633,

  np^0=60(0.633)=38>10

  Substitute n as 60 and q^0 as 0.367(=1p^0),

  nq^0=60(0.367)=22.02>10

Therefore, all the conditions are satisfied.

c.

To determine

Find the value of test statistic.

c.

Expert Solution
Check Mark

Answer to Problem 10E

The value of test statistic is –1.13.

Explanation of Solution

Calculation:

Denote p as the true population proportion.

The given test hypotheses are:

Null hypothesis:

 H0:p=0.7.

That is, the true proportion of the individuals under the category of interest is 0.7.

Alternate hypothesis:

H1:p0.7.

That is, the true proportion of the individuals under the category of interest is different from 0.7.

Test statistic:

The z-test statistic is:

z=p^p0p0(1p0)n,

Where, p^ be the sample proportion, p0 be the population proportion and n be the sample size.

Software procedure:

Step by step procedure to find the test statistic using the MINITAB software:

  • Choose Stat > Basic Statistics > 1-Proportion.
  • In Number of trials, enter the Sample size as 60 and in Number of Events, Enter Number of observations as 38.
  • Enter Hypothesized proportion as 0.7.
  • Select Options.
  • Choose Proportionhypothesized proprotion in Alternative hypothesis.
  • Click OK in all dialogue boxes.

The output using Minitab is given below:

Essential Statistics, Chapter 8.4, Problem 10E

From the MINITAB output, the test statistic, that is, the z-value is –1.13.

Thus, the value of test statistic z is –1.13.

d.

To determine

Decide whether the null hypothesis H0 is rejected at α=0.05 level.

d.

Expert Solution
Check Mark

Answer to Problem 10E

The null hypothesis H0 is not rejected at α=0.05 level.

Explanation of Solution

From previous part (c), it has been found that the value of test statistic z is –1.13.

From the given hypothesis, the alternative hypothesis contains the not equal () symbol. Thus, it is clear that the hypothesis follows the two-tailed test.

From Table 8.1 “Table of Critical Values”, the critical value for two-tailed test at α=0.05 level is ±1.96.

Therefore, the critical region is z1.96orz1.96.

Decision based on the critical value method:

For left-tailed test:  If zzα, reject H0.

For right-tailed test:  If zzα, reject H0.

For two-tailed test:  If zzα2orzzα2, reject H0.

Conclusion:

The critical value at α=0.05 level is ±1.96 and the test statistic is –1.13.

Here, the test statistic value of –2.80 lies within the critical region of ±1.96.

That is, zα2(1.96)<1.13<zα2(1.96).

Therefore, the null hypothesis is not rejected.

Hence, there is no evidence that the true proportion of the individuals under the category of interest is 0.7.

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Chapter 8 Solutions

Essential Statistics

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