In a packing plant, a machine packs cartons with jars. It is supposed that a new

Question

In a packing plant, a machine packs cartons with jars. It is supposed that a new machine will pack faster on the average than the machine currently used. To test that hypothesis, the times it takes each machine to pack ten cartons are recorded. The results, in seconds, are shown below: New Machine: 42.1 41.0 41.3 41.8 42.4 42.8 43.2 42.3 41.8 42.7 Old Machine:42.7 43.8 42.5 43.1 44.0 43.6 43.3 43.5 41.7 44.1 Let mu_O O be the population mean time it takes the old machine mu_N N be the population mean time it takes the new machine What is the appropriate test? What is the appropriate Alternative Hypothesis (HA)? HA: mu_O-mu_N=0 HA: mu_O-mu_N>0 HA: mu_O-mu_N<0 What is the value of the test statistic? Round to 2 decimal places. What is the p-Value of your test statistic? Round to 4 decimal places. Your decision is

Explanation / Answer

Let denote the mean for the new machine and denote the mean for the old machine.

Step 1. Ho: - = 0, Ha: - < 0

Step 2. Significance level: = 0.05.

Step 3. Compute the t-statistic:

Step 4. Critical value:

Left-tailed test
Critical value = - = - t0.05
Degrees of freedom = 10 + 10 - 2 = 18
- t0.05 = -1.734
Rejection region t < -1.734

Step 5. Check to see if the value of the test statistic falls in the rejection region and decide whether to reject Ho.

t* = -3.40 < -1.734
Reject Ho at = 0.05

Step 6. State the conclusion in words.

At 5% level of significance, the data provide sufficient evidence that the new machine packs faster than the old machine on average.

Question #2

Continuing from the previous example, give a 99% confidence interval for the difference between the mean time it takes the new machine to pack ten cartons and the mean time it takes the present machine to pack ten cartons.

Step 1. = 0.01, = t0.005 = 2.878, where the degrees of freedom are 18.

Step 2. The 99% confidence interval is (-2.01, -0.17).

Interpret the above result:

We are 99% confident that - is between -2.01 and -0.17.

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