Answered Essay: Recall that a bank manager has developed a new system to reduce the time customers spend

Recall that a bank manager has developed a new system to reduce the time customers spend waiting to be served by tellers during peak business hours. The mean waiting time during peak business hours under the current system is roughly 9 to 10 minutes. The bank manager hopes that the new system will have a mean waiting time that is less than six minutes. The mean of the sample of 90 bank customer waiting times is x = 5.43. If we let μ denote the mean of all possible bank customer waiting times using the new system and assume that σ equals 2.49; (a) Calculate 95 percent and 99 percent confidence intervals for μ (Round your answers to 3 decimal places.) 95 percent confidence intervals for μ is 99 percent confidence intervals for μ is (b) Using the 95 percent confidence interval, can the bank manager be 95 percent confident that u is less than six minutes?Explain Yes , 95 percent interval is below 6. c Using the 99 percent confidence interval, can the bank manager be 99 percent confident a μ s less than six minutes? Explain No ,99 percent interval extends above 6. (d) Based on your answers to parts b and c, how convinced are you that the new mean waiting time is less than six minutes? Fairly confident, since 95 percent Cl is below 6 while 99 percent Cl contains 6.

Recall that a bank manager has developed a new system to reduce the time customers spend waiting to be served by tellers during peak business hours. The mean waiting time during peak business hours under the current system is roughly 9 to 10 minutes. The bank manager hopes that the new system will have a mean waiting time that is less than six minutes. The mean of the sample of 90 bank customer waiting times is x = 5.43. If we let mu denote the mean of all possible bank customer waiting times using the new system and assume that sigma equals 2.49: (a) Calculate 95 percent and 99 percent confidence intervals for mu. (Round your answers to 3 decimal places.) (b) Using the 95 percent confidence interval, can the bank manager be 95 percent confident that mu is less than six minutes? Explain. 95 percent interval is 6. (c) Using the 99 percent confidence interval, can the bank manager be 99 percent confident that mu is less than six minutes? Explain. 99 percent interval extends 6. (d) Based on your answers to parts b and c, how convinced are you that the new mean waiting time is less than six minutes? confident, since 95 percent Cl is 6 while 99 percent Cl contains 6.

Expert Answer

The selection of a confidence level for an interval determines the probability that the confidence interval produced will contain the true parameter value. These levels correspond to percentages of the area of the normal density curve. For example, a 95% confidence interval covers 95% of the normal curve — the probability of observing a value outside of this area is less than 0.05. Because the normal curve is symmetric, half of the area is in the left tail of the curve, and the other half of the area is in the right tail of the curve. Thus , area in each tail of the curve is equal to (1-C)/2.

For a 95% confidence interval, the area in each tail is equal to 0.05/2 = 0.025. This will be equivalent to Z values = ( – 1.96, + 1.96)

Similarly , for a 99 percent confidence interval , the area in each tail is equal to 0.01/2 = 0.005. This will be equal to Z values = ( – 2.58 , + 2.58)

Given,

Sample size = n = 90

Sample mean = xbar = 5.43

Sample standard deviation = sd = 2.49

Formula for Confidence interval range for a specific confidence level value will be equal to

= Xbar + / – Z x sd/square root ( n)

Therefore 95 percent confidence interval will be

= 5.43 +/- 1.96 x 2.49 / Square root ( 90)

= 5.43 + /-   1.96 x 2.49/9.486

= 5.43 + /- 0.514 ( i.e. 4.916     and 5.944 )

And, 99 percent confidence interval will be

= 5.43 +/- 2.58 x 2.49 / Square root ( 90)

= 5.43 + /-   2.58 x 2.49/9.486

= 5.43 + /- 0.677 ( i.e. 4.753 and 6.107 )

95 PERCENT CONFIDENCE INTERVAL IS = 4.916, 5.944
99 PERCENT CONFIDENCE INTERVAL IS = 4.753 , 6.107

Based on above data,

It is clearly evident that for 95 percent confidence interval, process mean will be less than 6 minutes

However, for 99 percent confidence interval process mean can be more than 6 minutes ( since highest value of the range is 6.107)

While , it is possible that for 99 percent confidence interval new mean can be more than 6 upto 6.107, which is slightly above 6. However otherwise , it is very likely that new mean will be less than 6. Hence answer is “FAIRLY CONFIDENT”

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