Hypothesis, CI, and Regression - Critical 13 SELECTED Multi part problems from 4A, 5A and 6A
ashitjha
questions_hypothesis_13q_from_4a_5a_6a.doc1. To decrease the amount of time it takes to deliver packages, a delivery company purchased computer software that finds the optimal route for making deliveries based upon the input of the package destinations. In the past the average amount of time it took to complete deliveries was 9.4 hours with a standard deviation of 0.8 hours. Since purchasing the software the average delivery time over twenty delivery days was 8.6 hours. At the 0.05 level of significance, test whether the software package has reduced the average time it takes to complete deliveries.
2. Over the past few years, banks have strongly promoted online banking and the paying of bills online. In 2003, about 50% of internet users paid bills online. In a recent survey of 250 internet users, 150 said that they pay bills online. At the 5% level of significance, test whether the proportion of internet users paying bills online is now more than 50%.
Assignment 6A
Total Marks = 63
Please show all your work. Some questions require you to use graphs, tree diagrams, formulas, and so on that are difficult or impossible to reproduce in word-processing programs, so you will haveto print the entire assignment and fill it out and mail it to your tutor. As an alternative to mailing, you might be able to photograph or scan your assignment and submit it through the appropriate drop box. If you do, make sure the material you are submitting is legible.
Question 1
A manufacturer of exercise bicycles is studying the relationship between the number of months an exercise bicycle has been owned and the length of time it was used last week. Data for a random sample of 7 owners are provided in the table below
|
Months Owned ( ) x |
Hours Used Last Week ( ) y |
|
14 |
6 |
|
8 |
8 |
|
5 |
7 |
|
3 |
9 |
|
10 |
6 |
|
4 |
8 |
|
12 |
7 |
4 marks a. Construct a scatter diagram for these data with “Months Owned” on the horizontal
(
)
x
axis, and “Hours Used Last Week” on the vertical(
)
y
axis.Note: Try to make relatively full use of the graph area.
2 marks b. Describe the general pattern of the relationship between the two variables.
11 marks c. Calculate the least squares regression line with hours used as the dependent variable and months owned as the independent variable.
Note: Express a and b to 4 decimal places of accuracy.
3 marks d. Plot the regression line on the scatter diagram you constructed in part (a).
Note: Show your calculations for
=
4
x
and=
12
x
.
10 marks e. Can it be concluded that the slope of the regression line is negative? Formulate and test the appropriate hypotheses at the 5% significance level. Use the critical value approach.
4 marks f. Construct a 95% confidence interval for B.
2 marks g. Interpret the numerical value of b in the sample regression line. What does it mean in the context of this question?
2 marks h. Use the equation of the regression line to predict hours used last week if an exercise bicycle has been owned for 11 months.
Note: Express your answer to 2 decimal places of accuracy.
Question 2
An ice cream vendor would like to predict daily sales of ice cream cones based on the temperature outdoors. Data for a random sample of 8 days are provided in the table below.
|
Temperature ((C) ( ) x |
Daily Sales (Litres) ( ) y |
|
6 |
135 |
|
8 |
50 |
|
12 |
125 |
|
16 |
100 |
|
18 |
200 |
|
22 |
150 |
|
28 |
250 |
|
28 |
175 |
The regression line with “daily sales” as the dependent variable and “temperature” as the independent variable has been calculated as:
=+
ˆ
55.2175.38597
yx
You may use the following sums of squares and cross products for the questions below.
|
= å 138 x |
= å 1185 y |
|
|
= 495.5 xx SS |
= 26446.875 yy SS |
= 2668.75 xy SS |
2 marks a. Interpret the value of bin the sample regression line. What does it mean in the context of this question?
2 marks b. Compute the linear correlation between “daily sales” and “temperature.”
Note: Express your answer to 4 decimal places of accuracy.
8 marks c. At the 5% significance level, can it be concluded that the correlation between daily sales and temperature is positive? Formulate and test the appropriate hypotheses. Use the critical value approach.
2 marks d. What percentage of the variation in daily sales is explained by its linear relationship with temperature?
11 marks e. Using the regression line and sums of squares and cross products provided above, construct a 90% prediction interval for daily sales when the outside temperature is 24(C.
Assignment 5A
Total Marks = 88
Please show all your work. Some questions require you to use graphs, tree diagrams, formulas, and so on that are difficult or impossible to reproduce in word-processing programs, so you will haveto print the entire assignment and fill it out and mail it to your tutor. As an alternative to mailing, you might be able to photograph or scan your assignment and submit it through the appropriate drop box. If you do, make sure the material you are submitting is legible.
Question 1
A researcher is comparing the reaction times of 10-year-olds who play video games and 10-year-olds who do not. Random samples from each group provided the results shown below.
|
|
Reaction times of 10-year-olds |
|
|
|
Players of Video Games |
Non-players of Video Games |
|
Sample size |
15 |
10 |
|
Mean reaction time (seconds) |
0.50 |
0.52 |
|
Standard deviation of reaction times (seconds) |
0.01 |
0.02 |
Assume that the reaction times for both populations are normally distributed with the same population standard deviation.
7 marks a. Construct a 98% confidence interval for the difference between the population means of reaction times.
Note: Express your answer to 5 decimal places of accuracy.
8 marks b. Using a 2.5% significance level, can the researcher conclude that the mean reaction time of players of video games is less than the mean reaction time of non-players of video games? Formulate and test the appropriate hypotheses. Use the critical value approach.
Question 2
Is there any difference between the proportion of Americans who wish they were rich and the proportion of Canadians who wish they were rich? A random sample of 1,000 Americans showed that 550 wish they were rich, while a random sample of 500 Canadians showed that 225 wished they were rich.
6 marks a. Find a 97.5% confidence interval for the difference between the population proportions who wish they were rich.
Note: Express your answer to 4 decimal places of accuracy.
9 marks b. Test, at the 1% level of significance, whether the two population proportions are significantly different. Formulate and test the appropriate hypotheses. Use the critical value approach.
Question 3
A motivational program was designed to increase the number of hours each student who took it studied per week. Eight randomly selected students were involved in the program. The table below shows the number of hours each student studied per week before the taking motivational program and after taking the program.
|
|
|
|
|
|
|
|
|
|
|
Before program |
8 |
12 |
10 |
12 |
5 |
10 |
10 |
12 |
|
After program |
12 |
11 |
14 |
15 |
8 |
15 |
9 |
6 |
11 marks At the 10% level of significance, can it be concluded the motivational program increased the average number of hours students studied per week? Formulate and test the appropriate hypotheses. Use the critical value approach.
Assume that the population of paired differences has a normal distribution.
Question 4
It has been hypothesized that the distribution of seasonal colds in Canada is as follows:
|
Season |
Percentage |
|
Fall |
35% |
|
Winter |
25% |
|
Spring |
30% |
|
Summer |
10% |
A random sample of 200 Canadian citizens provided the following results:
|
Season |
Observed Frequency |
|
Fall |
80 |
|
Winter |
40 |
|
Spring |
70 |
|
Summer |
10 |
10 marks Do the observed data contradict the hypothesis? Formulate and test the appropriate hypotheses at the 5% level of significance. Use the critical value approach.
Question 5
A large company is interested in knowing whether there is any relationship between a person’s age and that person’s opinion about a new wage incentive plan. A random sample of 100 employees was selected and cross-classified as shown below.
|
|
|
Opinion |
|
|
|
Opposed |
Neutral |
In Favour |
|
Under 35 |
5 |
5 |
10 |
|
35 to 55 |
10 |
30 |
10 |
|
Over 55 |
15 |
10 |
5 |
10 marks At the 1% level of significance, can it be concluded that there is a relationship between an employee’s age and an employee’s opinion about the new wage incentive plan? Formulate and test the appropriate hypotheses. Use the critical value approach.
Question 6
A random sample of 16 students showed that the variance in the number of hours they spend studying for a final exam was 25 hours2.
7 marks a. Construct a 95% confidence interval for the population variance and standard deviation of hours spent studying. Assume the population of hours spent studying is normally distributed.
8 marks b. Test, at the 5% level of significance, whether the population standard deviation of hours spent studying is less than 10. Use the critical value approach.
Assignment 4A
Total Marks = 75
Please show all your work. Some questions require you to use graphs, tree diagrams, formulas, and so on that are difficult or impossible to reproduce in word-processing programs, so you will haveto print the entire assignment and fill it out and mail it to your tutor. As an alternative to mailing, you might be able to photograph or scan your assignment and submit it through the appropriate drop box. If you do, make sure the material you are submitting is legible.
Question 8
A study conducted last year indicated that adults spent an average of 14 hours on household chores during the weekend. A research group wishes to determine if the average hours spent on household chores during the weekend this year is different from the 14 hours of last year. Consequently, the research group selects a random sample of 64 adults and finds the mean hours spent on household chores during the weekend to be 15.05 hours. The population standard deviation is known to be 3.0 hours.
8 marks a. Formulate and test the appropriate hypotheses using the
p
-value approach. What is thep
-value of the test? Would you reject the null hypothesis ata
=
0.01
? What would be your conclusion?2 marks b. Would you reject the null hypothesis at
a
=
0.001
? What would be your conclusion?1 mark c. How strong is the evidence against
0
H
? [Refer to Unit 4, Section 7, in the Study Guide.]Question 9
8 marks A psychologist believes that if classical music is played during a test, test scores will increase. In the past, the mean test score was 60. A random sample of 49 individuals wrote the test while classical music was in the background. The sample results showed a mean test score of 62 and a standard deviation of 6.3. Using a 5% level of significance, can the psychologist conclude that the mean test score has increased with classical music? Formulate and test the appropriate hypotheses. Use the critical value approach.
Question 10
8 marks A telephone company estimates that 30% of its customers are interested in high-speed Internet service. To test this estimate, a random sample of 100 customers was selected and the results showed that 25 were interested in high-speed Internet service. Can it be concluded that the percentage of customers interested in high-speed Internet service is different from 30%? Formulate and test the appropriate hypotheses at a 5% significance level. Use the critical value approach.
