(A) One of your employees has suggested that your company develop a new product.
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Question
(A) One of your employees has suggested that your company develop a new product. You decide to take a random sample of your customers and ask whether or not there is interest in the new product. The response is on a 1 to 5 scale with 1 indicating "definitely would not purchase"; 2, "probably would not purchase"; 3, "not sure"; 4, "probably would purchase"; and 5, "definitely would purchase." For an initial analysis, you will record the responses 1, 2, and 3 as "No" and 4 and 5 as "Yes."
Suppose that after reviewing the results of a previous survey, you proceeded with preliminary development of the product. Now you are at the stage where you need to decide whether or not to make a major investment to produce and market it. You will use another random sample of your customers, but now you want the margin of error to be smaller. What sample size would you use if you wanted the 95% margin of error to be 0.013 or less? (Round your answer up to the nearest whole number.)
(B) Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the miles per gallon were recorded each time the gas tank was filled, and the computer was then reset. In addition to the computer's calculations of miles per gallon, the driver also recorded the miles per gallon by dividing the miles driven by the number of gallons at each fill-up. The following data are the differences between the computer's and the driver's calculations for that random sample of 20 records. The driver wants to determine if these calculations are different. Assume that the standard deviation of a difference is
= 3.0.
4.0
7.5
0.6
1.6
3.7
4.5
8.0
2.2
4.6
3.0
4.4
0.4
3.0
1.1
1.1
6.0
2.1
3.3
0.6
4.2
H0: = 0 mpg; Ha: 0 mpg
Carry out the test. Give the P-value. (Round your answer to four decimal places.)
4.0
7.5
0.6
1.6
3.7
4.5
8.0
2.2
4.6
3.0
4.4
0.4
3.0
1.1
1.1
6.0
2.1
3.3
0.6
4.2
Explanation / Answer
Part a
We assume, the responses as ‘No’ and ‘Yes’ will be same.
Proportion of response as ‘Yes’ = 50% = 0.50
That is, p = 0.50, q = 1 – p = 1 – 0.50 = 0.50
We are given
Confidence level = 95%
Margin of error = E = 0.013
Critical Z value = 1.96
(by using z-table)
Sample size formula is given as below:
Sample size = n = (Z/E)^2*p*q
Sample size = n = (1.96/0.013)^2*0.50*0.50 = 5682.84
Required sample size = 5683
Part b
Here, we have to use one sample z test for population mean.
H0: µ = 0 Vs Ha: µ 0
This is a two tailed test.
We assume 5% level of significance. = 0.05
Test statistic formula is given as below:
Z = (Xbar - µ) / [/sqrt(n)]
We have
Xbar = 2.755 (by using sample data)
= 3.00
n = 20
Z = (2.755 – 0) / [3.00/sqrt(20)]
Z = 2.755/ 0.6708
Z = 4.1069
P-value = 0.0000
P-value < , reject H0
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