Question

In 2011, the industries with the most complaints to the Better Business Bureau were banks, cable and satellite television companies, collection agencies, cellular phone providers, and new car dealerships (USA Today, April 16, 2012). The results for a sample of 200 complaints are contained in the DATAfile named BBB. Click on the datafile logo to reference the data.

Category	Observed Frequency
Bank	26
Cable	44
Car	42
Cell	60
Collection	28
Total	200

b. Using 0.01, conduct a hypothesis test to determine if the probability of a complaint is the same for the five industries.

The test-statistic is

p-value=

c. Dropping the industry with the most complaints using 0.05, conduct a hypothesis test to determine if the probability of a complaint is the same for the remaining four industries.

The test-statistic is

p-value=

Answer 1

Solution

b)

Test statistic = 19

P-value = 0.0008

Reject the null hypothesis

c)

Test statistic = 7.43

P-value= 0.0594

Do not reject null hypothesis

Working

The value of χ2χ2 is obtained as follows:

Observed Frequency (fi)(fi)	Expected Frequency (ei)(ei)	fi−eifi−ei	(fi−ei)2ei(fi−ei)2ei
26	40	–14	4.9
44	40	4	0.4
42	40	2	0.1
60	40	20	10
28	40	–12	3.6
Total			χ2=19

Thus, the value of χ2χ2 is 19.

Degrees of freedom:

The degrees of freedom is df=k−1df=k−1, where k is the number of categories.

df=k−1=5−1=4df=k−1=5−1=4

Thus, the degree of freedom is 4.

Level of significance:

The given level of significance is α=0.01

p-value:

Software procedure:

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

• Open an EXCEL sheet and select cell A1.
• In cell A1, enter the formula =CHISQ.DIST.RT(19,4).
• Press Enter.

The output obtained using EXCEL is given below:

From the output, the p-value is 0.0008.

Rejection rule:

• If the p-value≤αp-value≤α, then reject the null hypothesis.
• Otherwise, do not reject the null hypothesis.

Conclusion:

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

That is, p-value(=0.0008)<α(=0.01)p-value(=0.0008)<α(=0.01)

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

Therefore, there is sufficient evidence to conclude that the probability of a complaint is not the same for the five industries.

c)

Null hypothesis:

H0:p1=p2=p3=p4 H0:p1=p2=p3=p4.

That is, all population proportions are equal for five museums.

Alternative hypothesis:

Ha:not all population proportions are equal Ha:not all population proportions are equal.

That is, not all population proportions are equal for the remaining four industries.

The formula for chi-square test statistic is as follows:

χ2=∑i(fi−ei)2eiχ2=∑i(fi−ei)2ei.

Here, eiei is the expected frequency and fifi is the observed frequency for i^th observation.

In this case, it is observed that the probability of a complaint is the same for each of the four industries. That is, 0.25 (=14)(=14).

The expected frequency is obtained as follows:

Expected frequency=Sum of actual frequencies× Probability value=(26+44+42+28)×0.25=140×0.25=35Expected frequency=Sum of actual frequencies× Probability value=(26+44+42+28)×0.25=140×0.25=35

The value of χ2χ2 is obtained as follows:

Observed Frequency (fi)(fi)	Expected Frequency (ei)(ei)	fi−eifi−ei	(fi−ei)2ei(fi−ei)2ei
26	35	–9	2.31
44	35	9	2.31
42	35	7	1.4
28	35	–7	1.4
Total			χ2=7.42χ2=7.43

Thus, the value of χ2χ2 is 7.43

Degrees of freedom:

The degrees of freedom is df=k−1df=k−1, where k is the number of categories.

df=k−1=4−1=3df=k−1=4−1=3

Thus, the degree of freedom is 3.

Level of significance:

The given level of significance is α=0.05α=0.05.

p-value:

Software procedure:

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

• Open an EXCEL sheet and select cell A1.
• In cell A1, enter the formula =CHISQ.DIST.RT(7.42,3).
• Press Enter.

The output using EXCEL is given below:

From the output, the p-value is 0.0594.

Rejection rule:

• If the p-value≤αp-value≤α, then reject the null hypothesis.
• Otherwise, do not reject the null hypothesis.

Conclusion:

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

That is, p-value(=0.0596)>α(=0.05)