Question

Question 1 Production Rates

A manufacturing company wishes to find out whether the productivity of its workforce has increased now that they have a new machine. The factory examined the records for a random sample of 8 hours over the past month. The hourly production rates for these 8 hours were

:           142      175     162     158     190     154     160     185

Suppose the production rate before the use of the new machine is 150 units per hour with a standard deviation of 5. Test at 5% if the new machine has increased workers’ productivity on average using

(a) p-value approach, and

(b) Critical value approach.

Note: you must show all 7 steps for each hypothesis test

Answer 1

Given the details of sample as the hourly production rates for these 8 hours were

:           142      175     162     158     190     154     160     185

so, n=8 the production rate before the use of the new machine is the population mean,=150 units per hour with a standard deviation of =5.

Since the population standard deviation is known hence we will use the Z test statistic to conduct the hypothesis test.

a) P-value approach:

In the P-value approach, we compare the P-value with the level of significance to test the hypothesis.

Step 1.

Based on the test requires the hypotheses are:

Step-2

Test Type:

Based on the hypothesis it will be a right-tailed test.

Step-3

Rejection region:

At 0.05 level of significance reject the Ho if P-value calculated is less than 0.05.

Step 4

Test Statistic:

The test statistic is calculated as:

To calculate this we need to find the sample mean which is calculated as:

Mean = (142 + 175 + 162 + 158 + 190 + 154 + 160 + 185)/8
= 1326/8
Mean = 165.75

Now the test statistic:

Step-5

P-value:

P-value is calculated using excel formula at Z score calculated above, the excel formula is =1-NORM.S.DIST(8.91,TRUE)

Since for normal distribution, the probability obtained is always from the left side of the bell curve so, for a right tail test, we need to subtract it with 1.

Using excel formula we get a p-value less than 0.000001.

Step-6

Decision:

Since P-value is less than we reject the null hypothesis.

Step-7

Conclusion:

SInce P-value is less than the level of significance we can reject the null hypothesis and conclude that there is enough evidence to support the claim that the mean has increased of worker's productivity.

b) Critical value approach:

In a critical value approach, we compare the calculated test statistic with the critical value for critical region 0.05 which is level of significance.

Step 1.

Based on the test requires the hypotheses are:

Step-2

Test Type:

Based on the hypothesis it will be a right-tailed test.

Step-3

Rejection region:

The Critical region for 0.05 level of significance is calculated by excel formula =NORM.S.INV(0.95) hence the critical value for critical region is 1.645 Now the rejection region would be :

Reject Ho if Z >Zc=1.645

Step 4

Test Statistic:

The test statistic is calculated as:

To calculate this we need to find the sample mean which is calculated as:

Mean = (142 + 175 + 162 + 158 + 190 + 154 + 160 + 185)/8
= 1326/8
Mean = 165.75

Now the test statistic:

Step-5

P-value:

P-value is calculated using excel formula at Z score calculated above, the excel formula is =1-NORM.S.DIST(8.91,TRUE)

Since for normal distribution, the probability obtained is always from the left side of the bell curve so, for a right tail test, we need to subtract it with 1.

Using excel formula we get a p-value less than 0.000001.

Step-6

Decision:

Since Z-calculated is greater than Zc =1.645 hence we reject the null hypothesis.

Step-7

Conclusion:

SInce Z-critical is less than the Calculated Z score hence we can reject the null hypothesis and conclude that there is enough evidence to support the claim that the mean has increased of worker's productivity.