Question

1. Recall that a bank manager has developed a new system to reduce the time customers spend waiting for teller service during peak hours. The manager hopes the new system will reduce waiting times from the current 9 to 10 minutes to less than 6 minutes.

Suppose the manager wishes to use the random sample of 100 waiting times to support the claim that the mean waiting time under the new system is shorter than six minutes.

1. Letting μ represent the mean waiting time under the new system, set up the null and alternative hypotheses needed if we wish to attempt to provide evidence supporting the claim that μ is shorter than six minutes.

2. The random sample of 100 waiting times yields a sample mean of 5.46 minutes. Assuming that the population standard deviation equals 2.47 minutes, use critical values to test H₀ versus H_a at each of α = .10, .05, .01, and .001.

3. Using the information in part b, calculate the p-value and use it to test H₀ versus H_a at each of α = .10, .05, .01, and .001.

4. How much evidence is there that the new system has reduced the mean waiting time to below six minutes?

Answer 1

Solution

NOTE

Final answers in the desired format are given below. Back-up Theory and details of calculations follow at the end.

Part (a)

Hypotheses:

Null H₀: µ = µ₀ = 6 Vs Alternative H_A: µ < 6 Answer 1

Part (b)

Test statistic:

Z = (√n)(Xbar - µ₀)/σ = - 2.1862 Answer 2

Where

n = sample size;

Xbar = sample average;

σ = known population standard deviation.

Critical Values

-1.28155. - 1.645, - 2.3263, - 3.0902 for α= 0.1,0.05,0.01,0.001 Answer 3

Part (c)

p-value = 0.014 Answer 4

Part (d)

Decision

H₀ is rejected at significance levels 0.1 and 0.05, but accepted at significance levels 0.01 and 0.001 Answer 5

Conclusion

At significance levels 0.1 and 0.05, there is evidence to conclude that thennew system ha sreduced the waiting time, but not at significance levels 0.01 and 0.001. Answer 6

DONE

Back-up Theory and details of calculations

Let X = waiting time in minutes

Let µ and σ be the mean and standard deviation of X.

Claim: The new system reduces the waiting time to less than 6 minutes.

Hypotheses:

Null H₀: µ = µ₀ =    Vs Alternative H_A: µ < µ₀

Test statistic:

Z = (√n)(Xbar - µ₀)/σ, where n = sample size; Xbar = sample average; σ = known population standard deviation.

Summary of Excel Calculations is given below:

Given, n	100
µ0	6
σ	2.47
Xbar	5.46
Zcal	-2.18623
Given α	0.1,0.05,0.01,0.001
Zcrit	-1.28155 - 1.645 -2.3263 - 3.0902
p-value	0.014399

Distribution, Significance level α, Critical Value and p-value

Under H₀, Z ~ N(0, 1)

Critical value = lower α% point of N(0, 1).

p-value = P(Z < Zcal)

Using Excel Functions, Statistical NORMINV   NORMSDIST Zcrit and p-value are found to be as shown in the above table.

Decision:

Since Zcal < Zcrit, or equivalently, since p-value < α. H₀ is rejected/accepted.

complete