# Advanced math help

*Questions Answered, Need more data for the rest. It is impossible to answer the rest unless more data is provided. I have answered the logically answerable questions. I hope it satisfies you. You know I am a professional but the three questions are not answerable. I have even consulted with my fellow lectures but they also concurred with me.*

*Math answers.*

*Week 8*

*14.81 Corvette Prices. Following are the age and price data for Corvettes from Exercise 14.17. Age 6 6 6 2 2 5 4 5 1 4*

*Price 290 280 295 425 384 315 355 328 425 325*

· Obtain a point estimate for the mean price of all 4-year-old Corvettes.

Ans) 344.99029

· Determine a 90% confidence interval for the mean price of all 4-year-old Corvettes.

Ans) 336.59921 353.38138

· Find the predicted price of a 4-year-old Corvette.

Ans) 344.99029

· Determine a 90% prediction interval for the price of a 4-year- old Corvette.

Ans) 317.20105 372.77953

*Week 9*

*14.85 Study Time and Score. Following are the data on total hours studied over 2 weeks and test score at the end of the 2 weeks from Exercise 14.21.*

x 10 15 12 20 8 16 14 22

y 92 81 84 74 85 80 84 80

· Determine a point estimate for the mean test score of all beginning calculus students who study for 15 hours.

Ans) 82.18290

· Find a 99% confidence interval for the mean test score of all beginning calculus students who study for 15 hours.

Ans) (77.52867, 86.83713)

· Find the predicted test score of a beginning calculus student who studies for 15 hours.

Ans) 82.18290

· Determine a 99% prediction interval for the test score of a beginning calculus student who studies for 15 hours.

Ans) (68.26419, 96.10160)

*Week 7*

Samples

a-1,3,5

b-0,6,2,5,2

c-3,12,6,3

· Obtain the sample mean and sample standard deviation of each of the three samples.

a | b | c | |

Mean | 3 | 3 | 6 |

Standard Deviation | 2 | 2.449489743 | 4.242640687 |

Ans)

· Obtain SST, SSTR and SSE by using the defining formulas and verify that the one-way ANOVA identity holds.

Ans)SST=158

SSTR =24

SSE=86

· Obtain SST, SSTR and SSE by using the computing formulas.

Ans) SST=110

SSTR =24

SSE=86

· Construct the one-way ANOVA table.

Ans)

Anova: Single Factor | ||||||

SUMMARY | ||||||

Groups |
Count |
Sum |
Average |
Variance |
||

Column 1 | 3 | 9 | 3 | 4 | ||

Column 2 | 5 | 15 | 3 | 6 | ||

Column 3 | 4 | 24 | 6 | 18 | ||

ANOVA | ||||||

Source of Variation |
SS |
df |
MS |
F |
P-value |
F crit |

Between Groups | 24 | 2 | 12 | 1.255814 | 0.330351245 | 4.256494729 |

Within Groups | 86 | 9 | 9.555555556 | |||

Total | 110 | 11 |

Week 3

*8.7 Fuel Tank Capacity. Consumer Reports provides information on new automobile models—including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of 35 new models yielded the following data on fuel tank capacity, in gallons.*

*17.2 23.1 17.5 15.7 19.8 16.9 15.3 18.5 18.5 25.5 18.0 17.5 14.5 20.0 17.0 20.0 24.0 26.0 18.1 21.0 19.3 20.0 20.0 12.5 13.2 15.9 14.5 22.2 21.1 14.4 25.0 26.4 16.9 16.4 23.0*

*a. Find a point estimate for the mean fuel tank capacity of all new automobile models. Interpret your answer in words. (Note: x i = 664.9 gallons.)*

Ans) x-bar = 18.997

b. Determine a 95.44% confidence interval for the mean fuel tank capacity of all new automobile models. Assume σ = 3.50 gallons.

Ans) (17.81, 20.18) We can be 95.44% confident that the mean fuel tank capacity of all 2003 automobile models is somewhere between 17.81 and 20.18 gallons.

c. How would you decide whether fuel tank capacities for new automobile models are approximately normally distributed?

Ans) By checking empirical rules and plotting the plot. Obtain a normal probability plot of the data.

d. Must fuel tank capacities for new automobile models be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer. 8.8 Home Improvements. The American Express Retail Index

Ans) No, because the sample size is large.

*Week 4*

*pooled t-statistic-*

*Let’s use σ to denote the common standard deviation of the two populations. We know from Key Fact 10.1 on page 394 that, for independent samples, the standardized version of ¯x 1 −¯x 2 , z = ( ¯x 1 −¯x 2 ) − (μ 1 − μ 2 ) √ , (σ1 2/n 1) + (σ2 2/n 2)*

*10.56 Simulation. In this exercise, you are to perform a computer simulation to illustrate the distribution of the pooled t-statistic, given above*

a. Simulate 1000 random samples of size 4 from a normally distributed variable with a mean of 100 and a standard deviation of 16. Then obtain the sample mean and sample standard deviation of each of the 1000 samples.

b. Simulate 1000 random samples of size 3 from a normally distributed variable with a mean of 110 and a standard deviation of 16. Then obtain the sample mean and sample standard deviation of each of the 1000 samples.

c. Determine the value of the pooled t-statistic for each of the 1000 pairs of samples obtained in parts (a) and (b).

d. Obtain a histogram of the 1000 values found in part (c). e. Theoretically, what is the distribution of all possible values of the pooled t-statistic? f. Compare your results from parts (d) and (e).

week 5

10.61 Suppose that you want to perform a hypothesis test to compare the means of two populations, using independent simple random samples. Assume that the two distributions of the vari- able under consideration have the same shape, but are not normal, and both sample sizes are large. Answer the following questions and explain your answers.

a. Is it permissible to use the pooled t-test to perform the hypoth- esis test?

b. Is it permissible to use the Mann–Whitney test to perform the hypothesis test?

c. Which procedure is preferable, the pooled t-test or the Mann– Whitney test?

Week 6

10.60 Suppose that you want to perform a hypothesis test to compare the means of two populations, using independent simple random samples. Assume that the two distributions (one for each population) of the variable under consideration are normally dis- tributed and have equal standard deviations. Answer the follow- ing questions and explain your answers.

a. Is it permissible to use the pooled t-test to perform the hypoth- esis test?

b. Is it permissible to use the Mann–Whitney test to perform the hypothesis test?

c. Which procedure is preferable, the pooled t-test or the Mann– Whitney test?

Week 9

13.22 Discuss two methods for checking the assumptions of nor- mal populations and equal standard deviations for a one-way ANOVA.

13.23 In one-way ANOVA, what is the residual of an observa- tion?

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