Question

A very busy and fancy restaurant records the wait time for each customer that comes in. Below are 12 customers’ wait times (in minutes):

51        60        59        72        80        83        54        66        61        81        66        62

1.The manager wants to know if there is evidence that the true mean wait time is greater than 1 hour. What are the null and alternative hypotheses to test the claim? Will this be a one-sided or two-sided test?

2.Assuming that σ = 20 minutes, what is the test statistic for testing the hypotheses?

3.What is the corresponding p-value?

4.Do you reject the null hypothesis based on a significance level of .05? What conclusion should you tell the manager?

5.In practice, we do not know the population standard deviation. What would be the test statistic for testing the same hypotheses from (1.)?

6.According to the table of t critical values and assuming a significance level of α = .05, what is the critical value for rejecting the null hypothesis? What conclusion would you make?

Answer 1

1. Ho : <= 60 min

Ha: 60 min

Null hypothesis states that the mean wait time for the customer is 60 min.

This will be right tailed test as the manager wants to check if the true mean time is greater than 60 min

2. Test statistcs

As poulation SD is assumed, we will find z statistics

Mean for the sample data = sum of all terms / no of terms = 795/ 12 = 66.25

3. P value for z = 1.083

P ( Z>1.083 ) = 1−P ( Z<1.083 ) = 1−0.8599 = 0.1401

P value = 0.1401

4. Conclusion

= 0.05

The z-critical value for a right-tailed test, for a significance level of α=0.05

zc=1.64

Since z stat ( 1.083) does not fall in the rejection area, we fail to reject the Null hypothesis.

Also p value ( 0.1401) is greater than , we fail to reject the Null hypothesis.

Hence, we do not have sufficient evidence to believe that the mean wait time for customer is more than 60 min

5. If we do not know the poplation deviation, we need to use the sample standard deviation.

data	data-mean	(data - mean)2
51	-15.25	232.5625
60	-6.25	39.0625
59	-7.25	52.5625
72	5.75	33.0625
80	13.75	189.0625
83	16.75	280.5625
54	-12.25	150.0625
66	-0.25	0.0625
61	-5.25	27.5625
81	14.75	217.5625
66	-0.25	0.0625
62	-4.25	18.0625

6. T critical and conclusion

The t-critical value for a right-tailed test, for a significance level of α=0.05

tc=1.796

As the t stat is in the rejection area, we reject the Null hypothesis.