Question

a.

We have these hypotheses:

H0: the amount of active ingredient in a pharmaceutical pill is 5 mg

H1: the amount is below 5 mg

We can take a random sample of 40 such pills and find the amount of active ingredient in them, and let the sample average be x-bar. We also know that the standard deviation of the amount of active ingredient is 0.3 mg.

If H0 is right, what is the (approximate) distribution of  ?

		Normal with mean zero, standard deviation 1
		Normal with mean 5, standard deviation (0.3 / square root of 40)
		Normal with mean 5, standard deviation 0.3
		binomial with n = 40, p = 0.3

b.

A supermarket claims that the average wait time at the checkout counter is less than 9 minutes.  Assume that we know that the standard deviation of wait times is 2.5 minutes.

Consider

H0: mu >= 9

H1: mu < 9

A random sample of 50 customers yielded an average wait time of 8.5 minutes.

What is the p-value for this data?

(Provide four decimal places)-----------------

c.  

A supermarket claims that the average wait time at the checkout counter is less than 8 minutes.  Assume that we know that the standard deviation of wait times is 2.5 minutes. We will test at a significance level of 10%.

Consider

H0: mu >= 8

H1: mu < 8

A random sample of 50 customers yielded an average wait time of 7.8 minutes.

What is the critical value for the Zstat (the Z-test statistic)?

(Provide two decimal places)

d.

A manager is looking at the number of sick days used by employees in a year.

H0: the average number is 8 or below

H1: the average is over 8

We know that the standard deviation of the number of sick days used by employees is 2, and we want to test at 10% significance level.

Say we took a random sample of 50 employees, and checked their records, and found that the average was 8.1

The manager figures that the critical value (z-sub-0.1) is 1.28.

What should be the decision?

		We have insufficient information to make a decision
		Reject H0
		Employees are abusing their sick days
		Keep H0

Answer 1

a) Normal with mean 5, standard deviation (0.3 / square root of 40)

b)

Ho :   µ =   9  
Ha :   µ <   9   (Left tail test)
          

population std dev ,    σ =    2.5000  
Sample Size ,   n =    50  
Sample Mean,    x̅ =   8.5000  
          
'   '   '  
          
Standard Error , SE = σ/√n =   2.5/√50=   0.3536  
Z-test statistic= (x̅ - µ )/SE =    (8.5-9)/0.3536=   -1.4142  
          

          
p-Value   =   0.0786   [ Excel formula =NORMSDIST(z) ]

c) critical z value, z* =       -1.28 [Excel formula =NORMSINV(α/no. of tails) ]

d)

population std dev ,    σ =    2.0000
Sample Size ,   n =    50
Sample Mean,    x̅ =   8.1000
      
'   '   '
      
Standard Error , SE = σ/√n =   2/√50=   0.2828
Z-test statistic= (x̅ - µ )/SE =    (8.1-8)/0.2828=   0.35

decision: test stat <1.28 , Fail to reject Ho

Keep H0