Exercise #2
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 director’s 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 director’s 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 is worth marketing nationwide, we need to evaluate if the 85% requirement for public interest and the $47.50 requirement for the average price are met.
First, let's analyze the percentage of the public expressing interest.
Out of the 50 potential consumers surveyed, 34 expressed interest in purchasing the product. To determine the percentage, we divide the number of interested consumers by the total number of consumers surveyed and multiply by 100.
Percentage of consumers interested = (Number of interested consumers / Total number of surveyed consumers) * 100
Percentage of consumers interested = (34 / 50) * 100
Percentage of consumers interested = 68%
The director's requirement of at least 85% interest is not met as only 68% of the surveyed consumers expressed interest.
Next, let's analyze the average price consumers are willing to pay.
We have the maximum prices that each interested consumer is willing to pay. To meet the requirement of at least $47.50 as the average price, we need to calculate the average of these maximum prices.
Average of maximum prices = Sum of all maximum prices / Number of interested consumers
Average of maximum prices = (47 + 62 + 46 + ... + 63) / 34
To calculate the sum of all maximum prices, we add all the values provided:
47 + 62 + 46 + ... + 63 = 1587
Substituting this value into the equation:
Average of maximum prices = 1587 / 34
Average of maximum prices ≈ 46.68
The average price consumers are willing to pay is approximately $46.68, which is below the director's requirement of $47.50.
Based on these findings, I would reply to the director's request for information by stating that the product should not be marketed nationwide. The 85% interest requirement and the $47.50 average price requirement are not met.
Now, let's calculate the sample size needed to estimate the highest price consumers are willing to pay with a 95% level of confidence and a maximum error of $2.00.
To calculate the sample size, we need to use the formula:
n = (Z^2 * (σ^2)) / E^2
Where:
n = sample size
Z = z-score for the desired level of confidence (in this case, 95% confidence level corresponds to a z-score of approximately 1.96)
σ = standard deviation of the population (unknown in this case, so we'll use a sample standard deviation as an estimate)
E = maximum error (in this case, $2.00)
To estimate the standard deviation, we can use the standard deviation of the maximum prices provided.
Standard deviation of the sample = √[ sum((maximum price - average of maximum prices)^2) / (n-1) ]
Standard deviation of the sample = √[ sum((x - 46.68)^2) / (34-1) ]
Substituting the given values into the equation:
Standard deviation of the sample ≈ √[ sum((x - 46.68)^2) / 33 ]
We have the maximum prices for each consumer, so we can calculate the sum of squares of the differences: (x - 46.68)^2.
Next, we can substitute this estimated standard deviation and the other given values into the sample size formula to calculate the required sample size.
To determine if the new product is worth marketing nationwide, we need to analyze two factors: the percentage of the public expressing desire for the product and the average cost they are willing to pay. We also need to consider a confidence level of 95% to ensure the reliability of our findings.
First, let's determine if at least 85% of the public exhibit an interest in purchasing the product. Out of the 50 potential consumers surveyed, 34 expressed an interest. To find the percentage, we divide 34 by 50 and multiply by 100:
Percentage = (34/50) * 100 = 68%
Since the percentage of potential consumers interested in purchasing the product is below the 85% threshold set by the director, the product may not meet the requirement.
Next, we need to determine the average cost that potential consumers are willing to pay. We have a list of maximum prices provided by the surveyed individuals. To find the average, we sum up all the prices and divide by the number of individuals:
Average Price = (Sum of all prices) / (Number of individuals)
Sum of all prices = 47 + 62 + 46 + ... + 54 + 42 + 47 + 63
= 1051
Number of individuals = 50
Average Price = 1051 / 50
= $21.02
The average price that potential consumers are willing to pay is $21.02, which is below the required minimum price of $47.50 set by the director.
Based on these findings, we cannot recommend marketing the product nationwide as it fails to meet both the percentage and average price requirements.
To determine the sample size required for 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 can use the formula for sample size calculation:
Sample Size = (Z^2 * σ^2) / E^2
where Z is the Z-score corresponding to the desired confidence level, σ is the population standard deviation (unknown in this case), and E is the desired maximum error.
In this case, we want a 95% confidence level, which corresponds to a Z-score of approximately 1.96. The desired maximum error is $2.00.
Sample Size = (1.96^2 * σ^2) / 2^2
= (3.8416 * σ^2) / 4
Since we don't have the population standard deviation, we cannot calculate the exact sample size. However, we can estimate it by assuming a reasonable value for the population standard deviation.
Assuming a population standard deviation of $10, we can calculate the sample size:
Sample Size = (3.8416 * 10^2) / 4
= 9.604
Therefore, we would need a sample size of at least 10 to maintain a 95% level of confidence and a maximum error of $2.00 in estimating the highest price consumers are willing to pay.