Question

In: Statistics and Probability

A researcher wishes to estimate the mean yearly tuition of the population of private universities throughout...

  1. A researcher wishes to estimate the mean yearly tuition of the population of private universities throughout the United States. A random and representative sample of universities of this type is to be selected to provide the required estimate. To provide estimates of the population mean tuition, what minimum number of universities will be sampled under the following conditions to provide the needed estimate?
  1. The estimate desired will need to be computed with 95% confidence to within ±$4,000 when it is felt that the population standard deviation in the tuition amounts is $9,000.
  1. The estimate desired will now need to be computed with 90% confidence to within ±$4,000 when it is felt that the population standard deviation in the tuition amounts is $9,000.

  1. The estimate desired will now need to be computed with 95% confidence to within ±$4,000 when it is felt that the population standard deviation in the tuition amounts is $10,000.

  1. The estimate desired will now need to be computed with 95% confidence to within ±$5,000 when it is felt that the population standard deviation in the tuition amounts is $9,000.

Be sure to provide explanations of your answers. In addition, be sure to comment on any differences you observe in your answers. Give reasons for these differences. Do not use only mathematical justifications as you explain any differences in your answers. Make all comparisons relative to the answer found in the first part of the problem.

Solutions

Expert Solution

We know, Margin of error =

where n = Number of universities sampled

Corresponding to 95% confidence interval, the critical z score = 1.96

Corresponding to 90% confidence interval, the critical z score = 1.645

(a) Given Margin of error = 4000 and CI = 95%

-> ≤ 4000

-> n ≥ 19.45

Thus, minimum number of universities to be sampled = 20

(b) This is similar to (a) with CI = 90%

-> ≤ 4000

-> n ≥ 13.7

Thus, minimum number of universities to be sampled = 14

Since the confidence interval has decreased compared to (a) with standard deviation being same, lesser number of samples can give the same Margin of error

(c) This is similar to (a) with Standard deviation = $10,000

-> ≤ 4000

-> n ≥ 24.01

Thus, minimum number of universities to be sampled = 25

Since the standard deviation has increased compared to (a), higher number of samples are needed to give the same margin of error

(d) This is similar to (a) with margin of error increased to $5,000

-> ≤ 5000

-> n ≥ 12.45

Thus, minimum number of universities to be sampled = 13

Since the margin of error is higher compared to (a), lesser number of samples are needed.


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