Question 1A. Confidence Intervals for Paired Samples

Does drinking a small amount of alcohol reduce reaction time noticeably? Sixteen randomly selected subjects were given a test in which they had to push a button in response to the appearance of an image on a screen. Their reaction times were measured. Then the subjects consumed enough alcohol to raise their blood alcohol level to 0.05%. (In most states, a person is not considered to be "under the influence" until the blood alcohol level reaches 0.08%.) They then took the reaction time test again. Their reaction times, in milliseconds, are presented in the table below. Assume that the differences in reaction time are normally distributed. Construct a 95% confidence interval estimate of the mean difference in reaction time.

a = reaction times (in milliseconds) when blood alcohol level was 0.05%

b = reaction times (in milliseconds) when blood alcohol level was 0

a 102 100 77 61 85 50 95 115 64 98 107 44 47 92 70 94

b 103 99 69 50 96 26 71 109 53 89 103 27 50 100 66 86

d=a-b -1 1 8 11 -11 24 24 6 11 9 4 17 -3 -8 4 8

Step 1: Parameter & Population Random sampling? Normality? n > 30 or sample comes from a normal population.

n is the number of paired differences in the sample =

= mean(d) is the mean of the sample differences =

s = stdev(d) is the sample standard deviation of sample differences =

Steps 2 & 3: Critical Value & Margin of Error

(Use the T-table... df tells you what row to look at AND CL tells you which column)

Critical Value c = -------------- Margin of Error, E =------------------

Step 4: Confidence Interval

Calculations: ( - E, + E) = ------,-------

Is 0 in the interval?_____ Is the interval entirely negative?______ Is the interval entirely positive?_______

Step 5: Interpretation

Use this sentence frame "We are ___% confident that the true mean difference of __________________ ___________________________________________________ is between _________ and ____________.

Then choose one of the sentence starters to finish.

Since 0 is in the interval...

Since the interval is entirely positive...

Since the interval is entirely negative...

Question1B. Hypothesis Tests for Paired Samples

Does tuning a car engine improve the gas mileage? A random sample of eight automobiles were run to determine their mileage, in miles per gallon. Then each car was given a tune-up, and run again to measure the mileage a second time. The results are presented in the table below. Assume that the differences in gas mileage are normally distributed. We would like to determine how strong the evidence is that the population mean mileage is higher after tune-up.

Using a 1% level of significance, test the claim that the mileage is higher after the tune up.

a = after tune-up 35.44 35.17 31.07 31.57 26.48 23.11 25.18 32.39

b = before tune-up 33.76 34.30 29.55 30.90 24.92 21.78 24.30 31.25

difference d=a-b 1.68 0.87 1.52 0.67 1.56 1.33 0.88 1.14

Step 1: Determine the hypotheses

Define in words: "The mean difference of ......

H0 : = 0 (assuming there is no difference)

Ha : 0 (use <, > or depending on the claim)

Significance level = Right-tailed, Left-tailed or Two-tailed Test?

Step 2: Collect the data

Random sampling? Normality? n >30 or that sample comes from a normal population.

n is the number of paired differences in the sample =

= mean(d) is the mean of the sample differences =

s = stdev(d) is the sample standard deviation of sample differences =

Step 3: Assess the evidence

Test statistic: =_________________=____________

Use desmos to find the p-value P-value=__________ Graph with shading

Step 4: State a conclusion

Compare the p-value with the level of significance . What decisions should we make about H0 and Ha?

Explain what your decision means in the context of this problem