Question

In an article in the Journal of Marketing, Bayus studied the differences between "early replacement buyers" and "late replacement buyers" in making consumer durable good replacement purchases. Early replacement buyers are consumers who replace a product during the early part of its lifetime, while late replacement buyers make replacement purchases late in the product's lifetime. In particular, Bayus studied automobile replacement purchases. Consumers who traded in cars with ages of zero to three years and mileages of no more than 35,000 miles were classified as early replacement buyers. Consumers who traded in cars with ages of seven or more years and mileages of more than 73,000 miles were classified as late replacement buyers. Bayus compared the two groups of buyers with respect to demographic variables such as income, education, age, and so forth. He also compared the two groups with respect to the amount of search activity in the replacement purchase process. Variables compared included the number of dealers visited, the time spent gathering information, and the time spent visiting dealers. Regard the sample of 500 late replacement buyers for which σ = .58. How large a sample of late replacement buyers is needed to make us (Round up your answers to the next whole number.):

(a) 99 percent confident that ¯ x¯ , the sample mean number of dealers visited, is within a margin of error of .04 of µ, the population mean number of dealers visited? n______ buyers

(b) 99.73 percent confident that ¯ x¯ is within a margin of error of .05 of µ? n_____ buyers

Answer 1

(a)

The margin of error, E, for the population mean, , is computed using the formula:

where, is the right tail critical value corresponding to level of significance, is the population standard deviation and n is the sample size.

Here, . The left tailed critical value is found using the formula '=NORMINV()' in Excel. The screenshot is shown below:

This implies . Therefore, the right tail value is found below:

Now, in order to get the value of n, substitute the values of  , and in the formula stated above, as shown here:

Simplifying the above expression, the value of n, is obtained as:

Now, squaring both side;

Hence, the reuired sample consists of 1395 buyers.

(b)

The margin of error, E, for the population mean, , is computed using the formula:

where, is the right tail critical value corresponding to level of significance, is the population standard deviation and n is the sample size.

Here, . The left tailed critical value is found using the formula '=NORMINV()' in Excel. The screenshot is shown below:

This implies . Therefore, the right tail value is found below:

Now, in order to get the value of n, substitute the values of  , and in the formula stated above, as shown here:

Simplifying the above expression, the value of n, is obtained as:

Now, squaring both side;

Hence, the reuired sample consists of 1211 buyers.