As a marketing analyst for Imperial Products, your task is to determine if a new product is worth marketing nationwide. The director of marketing has stated that at least 85percent of the public must express some desire for the product and, for those who exhibit an interest, the average cost they are willing to pay must be at least $47.50. The director insists that you be 95 percent certain of your findings.

To respond to the directorƒ­s request, you survey 50 potential consumers regarding their interest in the product. Thirty-four express an interest in purchasing it. The maximum price each would be willing to pay is shown below. How would you reply to the directorƒ­s request for information? Should the product be marketed? How large should the sample be if you want to maintain a 95 percent level of confidence and suffer a maximum error of $2.00 in your estimate of the highest price consumers are willing to pay?
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Maximum Prices
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47 62 46
43 53 34
54 45 35
65 36 32
54 45 31
34 36 25
38 37 56
35 43 65
43 48 54
37 46 65
54 42 47
63
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To determine if the new product should be marketed nationwide, we need to assess two criteria set by the director of marketing: the percentage of the public expressing desire for the product and the average price they are willing to pay.

Let's analyze the data provided for the maximum prices consumers are willing to pay:

Maximum Prices:
47, 62, 46, 43, 53, 34, 54, 45, 35, 65, 36, 32, 54, 45, 31,
34, 36, 25, 38, 37, 56, 35, 43, 65,43, 48, 54, 37, 46, 65, 54, 42, 47, 63

Out of the 50 potential consumers surveyed, 34 expressed an interest in purchasing the product.

Now, let's calculate the percentage of the public expressing desire for the product:

Percentage = (Number of consumers interested / Total number of consumers) * 100

Percentage = (34 / 50) * 100 = 68%

Since 68% is less than the required threshold of 85%, we cannot conclude that the desired percentage of the public has expressed interest in the product. Therefore, based on this criteria alone, the product should NOT be marketed nationwide.

Next, let's calculate the average price consumers are willing to pay:

Average price = Total of maximum prices / Number of consumers interested

Average price = (47 + 62 + 46 + 43 + 53 + 34 + 54 + 45 + 35 + 65 + 36 + 32 + 54 + 45 + 31 + 34 + 36 + 25 + 38 + 37 + 56 + 35 + 43 + 65 + 43 + 48 + 54 + 37 + 46 + 65 + 54 + 42 + 47 + 63) / 34

Average price = 1379 / 34 = approximately $40.56

Since the average price consumers are willing to pay ($40.56) is less than the required threshold of $47.50, we cannot conclude that the average price requirement has been met. Thus, based on this criteria alone, the product should NOT be marketed nationwide.

To determine the sample size needed to maintain a 95% level of confidence and a maximum error of $2.00 in the estimate of the highest price consumers are willing to pay, we need more information. Specifically, we need to know the standard deviation of the population.

Please provide the standard deviation of the population or any additional information required to calculate the appropriate sample size.

To determine if the new product is worth marketing nationwide, we need to consider two criteria: the percentage of the public expressing an interest in the product and the average cost they are willing to pay.

First, let's calculate if at least 85% of the public expresses some desire for the product. In this case, out of the 50 potential consumers surveyed, 34 expressed an interest in purchasing it. So, to find the percentage, we divide the number of people interested (34) by the total number of people surveyed (50) and multiply by 100:

Interest percentage = (34/50) * 100 = 68%

Since 68% is less than 85%, it does not meet the director's requirement of at least 85% expressing an interest. Therefore, based on this criterion alone, we should not market the product nationwide.

Next, let's analyze the second criterion - the average cost consumers are willing to pay. The maximum prices consumers are willing to pay are listed below the question. To calculate the average, we add up all the maximum prices and divide by the number of responses:

Average cost = (Sum of all maximum prices) / (Number of responses)

Using the values provided, the sum of all maximum prices is 1275, and the number of responses is 34. Calculating:

Average cost = 1275 / 34 = $37.50

Since the average cost consumers are willing to pay ($37.50) is less than the required minimum of $47.50, the product does not meet the average cost criterion either.

Based on the results for both criteria, it is recommended not to market the product nationwide.

For the second part of your question, to determine the sample size needed to maintain a 95% level of confidence and suffer a maximum error of $2.00 in the estimate of the highest price consumers are willing to pay, we need to calculate the required sample size using a formula for sample size determination in confidence interval estimation.

The formula is:
n = [Z^2 * (σ^2)] / E^2

Where:
n = sample size
Z = Z-score associated with the desired level of confidence (95% corresponds to a Z-score of approximately 1.96)
σ = population standard deviation (unknown)
E = maximum error allowed in the estimate ($2.00)

Since we don't have the population standard deviation (σ), we would need to estimate it from a preliminary sample. However, since there is no information about the population standard deviation, we cannot calculate the exact sample size in this case. It is recommended to conduct a preliminary sample to estimate the standard deviation and then use that value to calculate the required sample size using the formula mentioned above.