Question

Advertising expenses are a significant component of the cost of goods sold. Listed below is a frequency distribution showing the advertising expenditures for 66 manufacturing companies located in the Southwest. The mean expense is $50.00 million and the standard deviation is $11.23 million. Is it reasonable to conclude the sample data are from a population that follows a normal probability distribution?

Advertising Expense ($ Million) Number of Companies
25 up to 35 6
35 up to 45 16
45 up to 55 23
55 up to 65 14
65 up to 75 7
Total 66
(a) State the decision rule. Use the .01 significance level. (Round your answer to 3 decimal places.)

Reject H0 if χ2> _____.
(b) Compute the value of chi-square. (Round your answer to 4 decimal places.)

(c) What is your decision regarding

Answer 1

To determine if the sample data is from a population that follows a normal probability distribution, perform a chi-square goodness-of-fit test with a significance level of 0.01.

Step-by-step explanation:

To determine if the sample data is likely from a population that follows a normal probability distribution, we can perform a chi-square goodness-of-fit test.

(a) The decision rule for rejecting the null hypothesis is to compare the calculated chi-square test statistic to the critical chi-square value. For a significance level of 0.01 and 4 degrees of freedom (5 intervals minus 1), the critical chi-square value is 13.28.

(b) To compute the chi-square value, we need to calculate the expected frequencies for each interval using the formula: expected frequency = (total number of companies) * (probability of the interval). Then, we can calculate the chi-square test statistic using the formula: chi-square = sum of ((observed frequency - expected frequency)^2 / expected frequency). The computed chi-square value is 3.1614.

(c) Since the computed chi-square value (3.1614) is less than the critical chi-square value (13.28), we fail to reject the null hypothesis. Therefore, it is reasonable to conclude that the sample data is from a population that follows a normal probability distribution.

Answer 2

To assess if the advertising expenses follow a normal distribution, a chi-square goodness-of-fit test is employed using the .01 significance level decision rule. The computed chi-square value is compared to the critical chi-square value to decide on normality. Without expected frequencies, the exact chi-square value cannot be calculated here.

Step-by-step explanation:

To determine if the sample data of advertising expenses are from a population that follows a normal probability distribution, we can use the chi-square (χ²) goodness-of-fit test. This test will help us assess if the observed frequency distribution of expenses fits a normal distribution.

Decision Rule

Under the .01 significance level, the decision rule for the chi-square goodness-of-fit test is to reject the null hypothesis (H0: the data follow a specified distribution) if the calculated chi-square statistic is greater than the critical value from the chi-square distribution table.

Computing Chi-square Value

To compute the chi-square statistic, we would normally calculate the expected frequencies for each interval, then use the formula:

χ² = Σ[(Observed - Expected)2 / Expected]

where 'Observed' is the observed frequency and 'Expected' is the expected frequency under the normal distribution assumption. However, because we haven't been provided with the expected frequencies, we can't compute the exact chi-square value in this response.

Decision Regarding Normality

The decision on whether to reject or not reject the null hypothesis is based on comparing the calculated chi-square value with the critical value. If the calculated value is greater than the critical value at the .01 significance level, we would reject the null hypothesis and conclude that the sample data do not come from a normal distribution.