Problem Scenario 1: write out 7 steps for all hypothesis tests

The following is a problem description. Use the mean & variance provided to answer the questions which follow, showing all calculations BY HAND. For all hypothesis tests, you MUST show all 7 steps we have been using and use the CRITICAL VALUE METHOD to make your decision.  Treat each question as a separate problem — we use the same data set but are answering different research questions.

Fitness Training
There are different approaches to fitness training. To judge which one of two approaches is better, 200 twenty-five-year-old men are randomly selected to participate in an experiment. For four weeks, 100 men are trained by Approach 1 while the other 100 men are trained by Approach 2.  The improvement in fitness was measured for each man and the statistics shown below were computed. The measurements are known to be normally distributed.

NOTE: use ONLY the Critical Value method for hypothesis tests. If you include both rules in step 4 or include both in your decision step, I will have to conclude that you do not yet understand them.

Approach 1      Approach 2
x _1=27.3      x _2=33.6
s_1^2=47.614      s_2^2=28.09

Answer the following four questions:

a) Estimate with 95% confidence the mean fitness improvement with Approach 2. Interpret the interval.

b) Do these results allow us to conclude at the 10% significance level that Approach 2 is superior to Approach 1 in improving fitness?

c) Scientists are interested in determining which of the two approaches results in more consistent fitness improvement. Do these results allow us to conclude at the 5% significance level that Approach 2 results in a more consistent improvement in fitness than approach 1?

d) Estimate with 99% confidence the variance of the fitness improvement with Approach 1. Interpret the interval.

Problem Scenario 2:

The following is a problem description. For all hypothesis tests, you MUST show all 7 steps we have been using and use the P-VALUE METHOD to make your decision.  For confidence intervals, there are not specific steps, but there is a specific Excel tool for each interval. Treat each question as a separate problem — we use the same data set but are answering different research questions.

Many parts of cars are mechanically tested to be certain that they do not fail prematurely. In an experiment to determine which one of two types of metal alloy produces superior door hinges, 40 of each type were tested until they failed. To evaluate how long hinges made with the different alloys would last, the number of openings and closings was observed and recorded (to the closest 0.1 million). Car manufacturers consider any hinge that does not survive 1 million openings and closings to be a failure., A statistician has determined that the number of openings and closings is normally distributed.

NOTE: use ONLY the P-value method for hypothesis tests. If you include both rules in step 4 or include both in your decision step, I will have to conclude that you do not yet understand the p-value rule.

The data is shown below, and the file is provided in canvas.

Number of Openings and Closings
    Alloy 1                                  Alloy 2
1.5    1.5    0.9    1.3        1.4    0.9    1.3    0.8
1.8    1.6    1.3    1.5            1.3    1.3    0.9    1.4
1.6    1.2    1.2    1.8            0.7    1.2    1.1    0.9
1.3    0.9    1.5    1.6            1.2    0.8    1.2    1.1
1.2    1.3    1.4    1.4            0.8    0.7    1.1    1.4
1.1    1.5    1.1    1.5            1.1    1.4    0.8    0.8
1.3    0.8    0.8    1.1            1.3    1.1    1.5    0.9
1.1    1.6    1.6    1.3            1.4    1.2    1.3    1.6
0.9    1.4    1.7    0.9            0.6    0.9    1.8    1.4
1.1    1.3    1.9    1.3            1.5    0.8    1.6    1.3

Answer the following four questions: write out 7 steps for all hypothesis tests

a) Can we conclude at the 5% significance level that the mean number of door openings and closings with hinges made from Alloy 1 is greater than 1.25 million?

b.) Can we conclude at the 10% significance level that the variance of the number of openings and closings for the hinges made from Alloy 2 is less than 0.065.

c.) Estimate with 90% confidence the difference in the number of openings and closings between hinges made with Alloy1 and hinges made with Alloy 2. Interpret the interval.

d.) The quality control manager is not only concerned about the openings and closings of the hinges but is also concerned about the proportion of hinges that fail. Can we infer at the 10% significance level that the proportion of hinges made with Alloy 2 that fail exceeds 18%?