TMC/F1874 Mathematics for Computing Assignment 1

1. A quality control engineer at a manufacturing plant is trying to determine the variation of parts that an operator is making with a particular machine. He takes a sample of n parts and for each part in the sample, measures how much the part width differs from the specifications for the part. He then uses the mean and standard deviation of these values to estimate the mean value for all parts the operator is making. Construct a 90% confidence interval for the mean value assuming a normal distribution and:

1. _n = 28, = 0.003, = 0.001
2. _n = 30, = 0.011, = 0.015

(11 marks)

2. A particular country contains 3850 mountain peaks. A sample of 36 of the country’s peak is studied, and the mean and standard deviation of the heights of these peaks are calculated. The sample mean is found to be 6240 ft. The boundaries of a 99% confidence interval for the mean height of all 3850 peaks are 6204.8 ft and 6275.2 ft. What is the standard deviation of the sample?

(5 marks)

3. A preliminary sample of 100 new children’s toys reveals that three of them are unsafe for children under five years old. A child advocacy group has mandated a 90% confidence interval for the percentage of all new toys that are safe. The group has also stated that the estimate for the percentage of all new toys that are safe must be within 0.025 of the actual value. What should the sample size to be to meet these requirements?

(3 marks)

4. Suppose a 90% confidence interval for turns out to be (1000, 2100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Explain how should you reduce the width of the confidence interval, and why.

(4 marks)

5. A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain’s new store in the mall. Fifteen credit card accounts were randomly selected and analyzed with the following results: = RM48.75,

= 361. Do you believe the average amount spent on the first visit to the chain’s new store in the mall exceeds RM35? Explain using a 90% confidence interval for µ.

(8 marks)

6. A health club manager claims that its members lose an average of 13 pounds during the first six months of membership. A sample of 15 members of this health club showed that they lost on average 11.5 pounds during the first six months of membership with a standard deviation of 2.15 pounds. Test at the 2.5% significance level if the mean weight lost during the first six month of their membership by all members of the health club is less than 13 pounds.

(8 marks)

7. The spokesman for the Ener-Go company, which manufactures an organic energy pills, has made the claim that his company’s product will, on average improve a runner’s time in the mile by at least 6.5 seconds. In order to test this claim, a group of 30 runners decides to compare their mile times before and after taking the pill. Describe the null and alternative hypothesis and the Type I and Type II errors for this test. Is this a right-tailed, left-tailed, or two-tailed test?

(5 marks)

8. A market analyst has estimated that 15% of American citizens who wear glasses or contact lenses purchase their eyewear through their personal optometrist, whereas the other 85% either don’t have a personal optometrist or have one but don’t purchase eyewear through them. The analyst conducts a random poll and asks 900 people where they purchase their eyewear. Results show that 107 of these people declare that they purchase their eyewear from the optometrist. Is the analyst’s estimation valid at the 0.01 level of significance? At the 0.05% level?

(6 marks)

9. (Data for analysis) Table I shows the examination marks (in %) for a total of 30 students meanwhile Table II shows the marks range and grade. You can also download the data in EXCEL under the name of ASSIGN_Data as attached. Answer the following questions using EXCEL. The screen shots of the output must be shown clearly.

Table I: Examination Mark for 30 students in Statistic Class

Table II: Marks Range & Grade

• Construct a Histrogram of frequency versus grade. (4 marks)

• Generate a descriptive Statistics report from Data Analysis Add-Ins. (2 marks)

• Construct a 95% confidence interval of mean examination marks from (b). (4 marks)

10. (Data for analysis) Suppose that you are working for an environmental group on household electricity consumption. You are primarily interested in quantifying what factors could affect the household electricity consumption. The environmental group provided a data set that contained the information of 50 randomly selected households in Western Sydney. The data aee recorded in an EXCEL file Assign_Data at Morpheus. Answer the following questions using EXCEL. The screen shots of the output must be shown clearly.

• The leader of the environmental group suspects that homes (referred to as population 1) consume more electricity than units (referred to as population 2). Perform a hypothesis test to validate such suspicion. You must clearly state your null and alternative hypothesis, your conclusions, and the related output that lead to your conclusions. Use α=0.05 and assume unequal variances. (8 marks)

• However, the test in Q1 does not take the size of the building into considerations. So you compute the electricity consumption per week per square meter. We call this the electricity consumption rate. Perform a hypothesis to verify if the electricity consumption rate of homes is higher than that of units. You must clearly state your null and alternative hypothesis, your conclusions, and the related output that lead to your conclusions. Use α=0.05 and assume unequal variances. (8 marks)
• Based on your analysis in Q10(a) and Q10(b), is it true that homes consume more electricity than units? (4 marks)