Question

In: Statistics and Probability

Advertising expenses are a significant component of the cost of goods sold. Listed below is a...

Advertising expenses are a significant component of the cost of goods sold. Listed below is a frequency distribution showing the advertising expenditures for 71 manufacturing companies located in the Southwest. The mean expense is $50.28 million and the standard deviation is $11.46 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                                                                 8

35 up to 45                                                                 12

45 up to 55                                                                    30

55 up to 65                                                                    12

65 up to 75                                                                     9

Total                                                                                 71

A.) State the decision rule. Use the 0.10 significance level. (round 2 decimal places)

H0: The population of advertising expenses follows a normal distribution

H1: The population of advertising expenses does not follow a normal distribution

B.) Compute the value of the chi-square. (round 2 decimal places)

Solutions

Expert Solution

Solution:-

For this kind of problem we can plot Histogram of Observed Frquency.

Here,

mean of expense is $50.28 million and standard deviation is $11.46 million

So,using this information we have to conclude that whether the data is actually coming from the normal population.Hence,we will be computing chi-square goodness of fit statistics and thereby conclude the findings.Null and alternative hypothesis for the above testing will be

H0: The population of advertising expenses follows a normal distribution

H1: The population of advertising expenses does not follow a normal distribution

A) Decision rule at 0.10 level of significance.

We will be rejecting the null hypothesis if,

" Chi-sq-calculated > Chi-sq-tabulated(alpha==0.10,df) "

or else

We can compare p-value of this test statistic with the level of significance(0.10)

If p-value > 0.10 then we reject the null hypothesis otherwise we do not sufficient evidence to reject the null hypothesis.

B) Compute the value of chi square.

Necessary formulas:-

We know that in case of the continuous distribution:-

And as E(X) = 50.28

sd(X) = 11.46

Hence it becomes-

Where Z is also standardized normal variable using mean and standard deviation.

So we will evaluate RHS probability using normal table.

Please find the calculations below.

Here,we can see that chi-calculated < chi-table;hence our decision will be to accept the null hypothesis.

Conclusion:- At 0.10 significance level, -We conclude that the data is coming from the normal population.

i.e "The population of advertising expenses follows a normal distribution" at 0.10 significance level.

Which we can see from below histogram of expected frequencies as well.

Both histograms are showing the normality of the frequencies.

orchestra answered 2 years ago
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