Question

. An electric car company’s sales manager is interested in the salaries of people who are on the wait list for their most affordable model and wonder if it is different from the median salary at their company of $135,000. They send a survey to a random sample of 2000 of the people on the waitlist and 85 of the respondents report having salaries above $135,000 and 103 have salaries below.

(a) Carry out and interpret a sign test of the hypothesis: Ho : M = $135, 000 vs HA : M 6= $135, 000 where M is the median salary of the people on the waitlist.

(b) Carry out and interpret a proportion test of the hypothesis: Ho : p = 0.5 vs Ho : p 6= 0.5, where p is the proportion of people on the waitlist with salaries above the median salary at the company.

(c) Why should the car company’s sales manager be cautious about putting too much weight on the results of this survey (hint: think about the data collection)?

Answer 1

a ) Sign test :

Hypothesis : Ho : M = $135, 000 vs HA : M $135, 000

                                                    where M is the median salary of the people on the waitlist.

So here X+ = number the respondents report having salaries above $135,000 = 85

            X- = number the respondents report having salaries below $135,000=103

X= Min (X+,X-)= Min(85,103) =85

Here n = Total number of sign = 85+103=188 > 25 is large enough to use normal approximation, so a z-statistic will be used.

Rejection Region

Based on the information provided, the significance level is , and the critical value for a two-tailed test is and

The rejection region for this two-tailed test is R= { test statitics < -1.96 ot test statistics > 1.96}.

Test Statistics

The z-statistic is computed as follows,

                 

So here observed test statistics value = - 1.24 > - 1.96 ;it is concluded that the null hypothesis is not rejected.

Conclusion:

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to the median salary at their company is different $135,000, at the 0.05 significance level.

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Part b ) .

A proportion test:

Hypothesis:      Ho : p = 0.5 vs Ho : p 0.5,

                                               where p is the proportion of people on the waitlist with salaries above the median salary at the company.

x = number of people on the waitlist with salaries above the median salary at the company. =85

n = number of the people on the waitlist in random sample = n = 2000.

p^ = x/n = 85/2000 = 0.0428.

This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.

Rejection Region :

Based on the information provided, the significance level is , and the critical value for a two-tailed test is and .

The rejection region for this two-tailed test is R= { test statitics < -1.96 ot test statistics > 1.96}.

Test Statistics

The z-statistic is computed as follows:

Decision about the null hypothesis:

So here test statistics = - 40.92 < -1.96 , it is then concluded that the null hypothesis is rejected.

Conclusion:

It is concluded that the null hypothesis Ho is rejected. Therefore, there is not enough evidence to the proportion of people on the waitlist with salaries above the median salary at the company.is 50% at the 5% significance level.

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Here here ant to , An electric car company’s sales manager is interested in the salaries of people who are on the wait list for their most affordable model and wonder if it is different from the median salary at their company of $135,000.

So here have to take care about , median salaries of people who are on the wait list should be not greater than the median salary at their company of $135,000..