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# Hypothesis Testing: Confidence Intervals

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7. In a packing plant a machine packs cartons with jars. Supposedly a new machine will pack faster on the average than the machine currently used. To test that hypothesis the time it takes each machine to pack 10 cartons are recorded. The results in seconds are shown in the following table.
New:
42,41,41.3,41.8,42.4,42.8,43.2,42.3,41.8,42.7
Old:
42.7,42.6,43.8,43.3,42.5,43.5,43.1,41.7,44,44.1
a.Determine the 95% interval for the difference between the mean time it takes the new machine to pack 10 cartons and the mean time it takes the present machine to pack two cartons.
b. interpret your result from part a
c. is there enough evidence to keep the present machine? Use alpha = 0.05
d. what assumptions are needed for the test to be valid?

9. Suppose it is known that 65% of adults in Michigan have a cell phone. You plan to take a simple random sample of 40 adults from Michigan
a. what is the probability that 75% or more of the adults in your sample will have a cell phone.
b. would it be quite unusual to find that 25% of the adult in your sample that have a cell phone give support for your answer.

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#### Solution Preview

Please see the attached file for fully formatted explanation.
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7. In a packing plant a machine packs cartons with jars. Supposedly a new machine will pack faster on the average than the machine currently used. To test that hypothesis the time it takes each machine to pack 10 cartons are recorded. The results in seconds are shown in the following table.
New:
42,41,41.3,41.8,42.4,42.8,43.2,42.3,41.8,42.7
Old:
42.7,42.6,43.8,43.3,42.5,43.5,43.1,41.7,44,44.1

a. Determine the 95% interval for the difference between the mean time it takes the new machine to pack 10 cartons and the mean time it takes the present machine to pack two cartons.
b. interpret your result from part a
c. is there enough evidence to keep the present machine? Use alpha = 0.05
d. what assumptions are needed for the test to be valid?

Solution: (see attached file).

Substituting the values we have,

After simplification we have ...

#### Solution Summary

This solution explains how to construct 95% confidence intervals in the context of different statistical problems.

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## One-Sample Hypothesis Test Confidence Intervals

One-Sample Hypothesis Test Confidence Intervals

Take a random sample and perform a t-test against a population mean (no time series for any parts of this project.) Test a hypothesis about a population mean. Besides the instructions above, explain what assumptions you made. Note: n must be between 15 to 20! Use no other sample sizes.

Use the same data as 1a and develop a 90% confidence interval and a 95% confidence interval for the population mean. Contrast this to 1a. What additional information or insight did this provide you over the traditional hypothesis testing.

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