Download It is determined that the mean vehicle miles traveled by a U.S.

1. It is determined that the mean vehicle miles traveled by a U.S. household annually
is greater than 22,000 miles. You conduct some research and determine that a
random sample of 36 U.S. households has a mean annual vehicle miles of travel of
22,000 with a standard deviation of 775 miles. You conduct a statistical experiment
where
H0: µ≤ 22,000 and Ha: µ>22,000. At α = 0.05, explain why you cannot reject H0.
Solution: (we use z distribution because n>30, we could use t statistic too)
H0: µ≤ 22,000
Ha: µ>22,000
n = 36, x-bar =22,000, s=775
Statistic = z = (x-bar-22,000)/(775/√36) = 0
Critical value is: z(0.05) = 1.645
Critical region = {z/z>1.645}
Answer: : We fail to reject Ho because the statistic value (0) is not greater than
1.645 (Statistic value is not in the critical region)
2. A local Jiffy Lube franchise questioned the frequency of oil changes. They
believed that people traveled more than 3500 miles between oil changes. They took
a random sample of 8 cars getting an oil change which has a mean distance of 3375
miles since having an oil change with a standard deviation of 225 miles. At α =
0.05, do you have enough evidence to support Jiffy Lube’s claim?
Solution:
We use t-statistic because n=8 (n<30)
Ho:  ≥ 3500
Ha:  < 3500
n=8, x-bar=3375, s=225, α=0.05
Statistic = t = (x-bar-3500)/(s/√n) = (3375-3500)/(225/√8) = -1.571
Critical value = -t(α,n-1) = -t(0.05,7) = -1.895
Critical region = {t/t<-1.895}
Decision: We fail to reject Ho since the ststistic value (-1.571) is not less than 1.895
Answer: We don`t have enough evidence to reject the claim ( ≥ 3500)
3. The table below shows the number of robberies reported (in millions) and the
number of convictions reported (in millions) by the U.S. department of justice for
14 years. At α= 0.05, can you conclude that there is a significant linear correlation
between the number of robberies and the number of convictions?
Robberies, x 1.60 1.55 1.44 1.40 1.32 1.23 1.22
Convictions, y 0.78 0.80 0.73 0.72 0.68 0.64 0.63
Robberies, x 1.23 1.22 1.18 1.16 1.19 1.21 1.20
Convictions, y 0.63 0.62 0.60 0.59 0.60 0.61 0.58
Perform a hypothesis test to make a conclusion about the indicated correlation
coefficient. Use the Table 5- t-distribution. 8 points
Solution:
Using excel coefficient of correlation = r = 0.981
Ho:  = 0
Ha:  ≠ 0
n = 14, r=0.981, α= 0.05
Statistic = t = r√(n-2)/√(1-r2) = 0.981√12/√(1-(0.981)2) = 17.516
Critical values are: -t(0.025,12) = -2.179 and t(0.025,12) =2.179
Critical region = {t/t<-2.179 or t>2.179}
Decision: we reject Ho since the statistic value (17.156) is less than -2.179
Interpretation: There is a significant linear correlation between the number of
robberies and the number of convictions
4. What does switching the explanatory and response variables have on the
correlation coefficient? Calculate the correlation coefficient, letting row 1 represent
x values and row 2 represent y values. Then calculate the correlation coefficient, r,
letting row 2 represent the x values and row 1 represent the y values.
Row 1 0 1 2 3 3 5 5 5 6 7
Row 2 96 85 82 74 95 68 76 84 58 65
Row 1 = x values
Row 2 = y values
r = -0.779
Row 1 = y values
Row 2 = x values
r = -0.779
Answer: correlation coefficient remains unchanged
5. For the given data, find the equation of the regression line letting row 1 represent
the x-values and row 2 the y-values. Sketch a scatter plot of the data and draw the
regression line. Then find the equation of the regression line letting row 1 represent
the y-values and row 2 the x-values. Sketch a scatter plot of the data and draw the
regression line. What effect does switching the explanatory and response variables
have on the regression line?
Row 1 16 25 39 45 49 64 70
Row 2 109 122 143 132 199 185 199
Row 1: x-values
Row 2: y values
Regression line: y = 1.7236x+79.733 (see graph)
250
y = 1.7236x + 79.733
200
150
Series1
Linear (Series1)
100
50
0
0
20
40
60
80
Row 1: y-values
Row 2: x values
Regression line: y = 0.4528x-26.448 (see graph)
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y = 0.4528x - 26.448
70
60
50
Series1
40
Linear (Series1)
30
20
10
0
0
50
100
150
200
250
Answer: Switching the variables make a change of the regression line