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# Linear Regression and R-Squared Analysis

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1. A random sample of size 36 from a normal population yields mean X-bar = 32.8 and standard deviation s = 4.51. Construct a 95 percent confidence interval for μ. (Ch9)
2. Test at α = 0.05, the hypotheses H0: µ= 0.33 versus HA: µ < 0.33 with p = 0.23 and n=100 (Ch10) (show the calculated and critical values of the test statistic)
3. A local tire dealer wants to predict the number of tires sold each month. He believes that the number of tires sold is a linear function of the amount of money invested in advertising. He randomly selects 6 months of data consisting of tire sales (in thousands of tires) and advertising expenditures (in thousands of dollars). Based on the data set with 6 observations, the simple linear regression model yielded the following results.

∑X = 24
∑X2 =124
∑Y = 42
∑Y2 = 338
∑XY = 196

Calculate the coefficient of determination and the coefficient of correlation between X and Y. Interpret the coefficient of Determination. Also find the slope and intercept and write the estimated Regression equation. What would the predicted sales of tires be if he spends five thousand dollars in advertising? (Ch13)
4. Suppose that the waiting time for a license plate renewal at a local office of a state motor vehicle department has been found to be normally distributed with a mean of 30 minutes and a standard deviation of 8 minutes. What is the probability that a randomly selected individual will have a waiting time between 15 and 45 minutes? (Ch7)
5. A microwave manufacturing company has just switched to a new automated production system. Unfortunately, the new machinery has been frequently failing and requiring repairs and service. The company has been able to provide its customers with a completion time of 6 days or less. To analyze whether the completion time has increased, the production manager took a sample of 25 jobs and found that the sample mean completion time was 6.5 days with a sample standard deviation of 1.5 days. At a significance level of 0.05, perform the test whether the completion time has increased. (Form the hypotheses, show the critical and calculated values of the test statistic and perform the test) (Ch10)
6. An experiment was performed on a certain metal to determine if the strength is a function of heating time. Results based on 15 metal sheets are given below. Use the simple linear regression model.
∑X = 30
∑X2 = 100
∑Y = 45
∑Y2 = 165
∑XY = 120

Find the slope and intercept and write the estimated Regression equation. Find SSE, SST, and SSR. Also find the Coefficient of Determination and interpret it. Find the correlation coefficient and check its relation with the Coefficient of Determination. Predict the value of Y when X is equal to 3. (Ch13)
7. In a survey of 1,000 people, 420 are opposed to the tax increase. Construct a 95 percent confidence interval for the proportion of those people opposed to the tax increase. (Ch9)
8. The diameter of small Nerf balls manufactured at a factory in China is expected to be approximately normally distributed with a mean of 5.2 inches and a standard deviation of 0.08 inches. Suppose a random sample of 20 balls is selected. What percentage of sample means will be less than 5.15 inches? (Ch8)
9. Container 1 has 8 items, 3 of which are defective. Container 2 has 5 items, 2 of which are defective. One item is drawn from each container: What is the probability that exactly one of the items drawn is defective? (Hint: you have to use both rule of addition and rule of multiplication/conditional probability rules) (Ch5)
10. An important part of the customer service responsibilities of a cable company relates to the speed with which trouble in service can be repaired. Historically, the data show that the likelihood is 0.75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that: Fewer than two troubles will be repaired on the same day? (Ch6)

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#### Solution Preview

1. A random sample of size 36 from a normal population yields mean X-bar = 32.8 and standard deviation s = 4.51. Construct a 95 percent confidence interval for μ. (Ch9)
The critical value for 95% confidence interval is 1.96.
Margin of error=1.96*4.51/sqrt(36)=1.47
Upper limit: 32.8+1.47=34.27
Lower limit: 32.8-1.47=31.33
Therefore, the 95% confidence interval is [31.33, 34.27].
2. Test at α = 0.05, the hypotheses H0: µ= 0.33 versus HA: µ < 0.33 with p = 0.23 and n=100 (Ch10) (show the calculated and critical values of the test statistic)
This is one tailed test. The degree of freedom is 100-1=99, at 0.05 significance level, the critical t value is -1.66.
Test value=(0.23-0.33)/sqrt(0.33*(1-0.33)/100)= -2.13
Since -2.13<-1.66, we could reject the null hypothsis.
Based on the test, we could conclude that µ < 0.33.
3. A local tire dealer wants to predict the number of tires sold each month. He believes that the number of tires sold is a linear function of the amount of money invested in advertising. He randomly selects 6 months of data consisting of tire sales (in thousands of tires) and advertising expenditures (in thousands of dollars). Based on the data set with 6 observations, the simple linear regression model yielded the following results.

∑X = 24
∑X2 =124
∑Y = 42
∑Y2 = 338
∑XY = 196

Calculate the coefficient of determination and the coefficient of correlation between X and Y. Interpret the coefficient of Determination. Also find the slope and intercept and write the estimated Regression equation. What would the predicted sales of tires be if he spends five thousand dollars in advertising? (Ch13)
SST=∑Y2 -(∑Y/6 )^2*6=338-(42/6)^2*6=44.
SSR=(
Slope m=((n∑XY-(∑X*∑Y ...

#### Solution Summary

This solution helps with a problem regarding linear regression and r squared analysis.

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