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Comparing Confidence Intervals

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1. The eruption height and the time interval after eruption of a geyser were measured and are shown below.
Height (x) 107 145 123 120 127 124 138 150
Interval after (y) 62 84 73 66 78 67 80 83

a. Find the value of the linear correlation coefficient r.
r= _____ (round to three decimal places as needed)

b. Find the critical value of r from the table showing the critical values for the Pearson correlation coefficient using a = 0.05
The critical values are + ________ (round to three decimal places as needed)

c. Is there sufficient evidence to conclude that there is a linear correlation between the two variables?
a. Yes, because the absolute value to the correlation coefficient is less than the critical value
b. Yes, because the absolute value of the correlation coefficient is greater than the critical value.
c. No, because the absolute value of the correlation coefficient is less than the critical value.
d. No, because the absolute value of the correlation coefficient is greater than the critical value.

2.
Use a scatterplot and the linear correlation coefficient r to determine whether there is a correlation between the two variables use a=0.05
X 2 4 4 1 3
Y 3 2 7 4 1

Does the given scatterplot suggest that there is a linear correlation?
a. Yes, because the data does not follow a straight line
b. No, because the points do not appear to have a straight line pattern
c. Yes, because the points appear to have a straight line pattern.
d. No, because the data follows a straight line.

Does the correlation coefficient indicate that there is a linear correlation between the variables?
a. No, because the absolute value of the correlation coefficient is greater than the critical value
b. No, because the absolute value of the correlation coefficient is less than the critical value
c. Yes, because the absolute value of the correlation coefficient is greater than the critical value
d. Yes, because the absolute value of the correlation coefficient is less than the critical value.

3. In a study designed to test the effectiveness of magnets for treating back pain, 40 patients were given at treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from ) (no pain) to 100 (extreme pain). After given the magnet treatment, the 40 patients had pain scores with a mean of 7.0 and a standard deviation of 2.3. After being given the sham treatments, the 40 patients had pain scores with a mean of 5.4 and a standard deviation of 2.1.

a. Construct the 95% confidence interval estimate of the mean pain score for patients given the magnet treatment
What is the confidence interval estimate of the population mean u?
____<u<_______ (round to one decimal place as needed)

b. Construct the 95% confidence interval estimate of the mean pain score for patients given the sham treatment
What is the confidence interval estimate of the population mean u?
______<u< _______ (round to one decimal place as needed)

c. Compare the results. Does the treatment with magnets appear to be effective?

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The solution compares confidence intervals and answers step-by-step various statistical questions.

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1. The eruption height and the time interval after eruption of a geyser were measured and are shown below.
Height (x) 107 145 123 120 127 124 138 150
Interval after (y) 62 84 73 66 78 67 80 83

a. Find the value of the linear correlation coefficient r.
r= _____ (round to three decimal places as needed)
mean of x: (107+145+...+150)/8=129.25

SSx=(107-129.25)^2+(145-129.25)^2+...+(150-129.25)^2=1407.5
Mean of y: (62+84+...+83)/8=74.125

SSy=(62-74.125)^2+(84-74.125)^2+...+(83-74.125)^2=490.875

Sxy=(107-129.5)*(62-74.125)+(145-129.25)*(84-74.125)+...+(150-129.25)*(83-74.125)= 771.75

R=Sxy/sqrt(SSx*SSy)=771.75/sqrt(1407.5*490.875)= 0.928468

b. Find the critical value of r from the table showing the critical values for the Pearson correlation coefficient using a = 0.05 , the degree of freedom is 8-2=6.
The critical values are + ____2.447____ (round to three decimal places as needed)
test value t=0.928468*sqrt((8-2)/(1-0.928468^2))=6.123

Since 6.123>2.447, we reject null hypothesis. Therefore, b is ...

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