TIM7 100
DATA FILE 5
ASSIGNMENT 5
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Order Paper Now- Independent random samples of 64 observations each are chosen from two normal populations with the following means and standard deviations:
Population 1 | Population 2 |
µ_{1} = 12 | µ_{2} = 10 |
Ϭ_{1} = 4 | Ϭ_{2} = 3 |
Let _{1} and _{2} denote the two sample means.
- Give the mean and standard deviation of the sampling distribution of _{1}.
- Give the mean and standard deviation of the sampling distribution of _{2}.
- Suppose you were to calculate the difference (_{1} – _{2}) between the sample means. Find the mean and standard deviation of the sampling distribution of (_{1} – _{2}).
- Will the statistic (_{1} – _{2}) be normally distributed? Explain.
- In order to compare the means of two populations, independent random samples of 400 observations are selected from each population, with the following results:
Sample 1 | Sample 2 |
_{1}= 5,275 | _{2} = 5,240 |
s_{1} = 150 | s_{2} = 200 |
- Use a 95% confidence interval to estimate the
difference between the population means (µ_{1} – µ_{2}). Interpret the confidence interval.
- Test the null hypothesis H_{0}: (µ_{1} – µ_{2}) = 0 versus the alternative hypothesis H_{a}: (µ_{1} – µ_{2}) ≠ 0. Give the significance level of the test, and interpret the result.
- Suppose the test in part b was conducted with the alternative hypothesis H_{a}: (µ_{1} – µ_{2}) > 0. How would your answer to part b change?
- Test the null hypothesis H_{0}: (µ_{1} – µ_{2}) = 25 versus the alternative hypothesis H_{a}: (µ_{1} – µ_{2}) ≠ 25. Give the significance level and interpret the result. Compare your answer to the test conducted in part b.
- What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a – d?.
- Assume that Ϭ_{1}^{2 }= Ϭ_{2}^{2}
= Ϭ^{2}. Calculate the pooled
estimator of Ϭ^{2} for each of the following cases:
- s_{1}^{2 }= 120, s_{2}^{2 }= 100, n_{1} = n_{2} = 25
- s_{1}^{2 }= 12, s_{2}^{2 }= 20, n_{1} =20, n_{2} = 10
- s_{1}^{2 }= 0.15, s_{2}^{2 }= 0.20, n_{1} =6, n_{2} = 10
- s_{1}^{2 }= 3,000, s_{2}^{2 }=2,500, n_{1} =16, n_{2} = 17
- Note that the pooled estimate is a weighted average of the sample variances. To which of the variances does the pooled estimate fall nearer in each of the above cases?
- Suppose you manage a plant that purifies its liquid waste and discharges the water into a local river. An EPA inspector has collected water specimens of the discharge of your plant and also water specimens in the river upstream from your plant. Each water specimen is divided into five parts, the bacteria count is read on each, and the mean count for each specimen is reported. The average bacteria count for each of six specimens are reported in the following table for the two locations.
Plant Discharge | Upstream | ||||
30.1 | 36.2 | 33.4 | 29.7 | 30.3 | 26.4 |
28.2 | 29.8 | 34.9 | 27.3 | 31.7 | 32.3 |
- Why might the bacteria counts shown here tend to
be approximately normally distributed?
- What are the appropriate null and alternative hypotheses to test whether the mean bacteria count for the plant discharge exceeds that for the upstream location? Be sure to define any symbols you use.
- What assumptions are necessary to ensure the validity of this test?
- A paired difference experiment produced the following data:
n_{D} = 18 _{1 }= 92 _{2 }= 95.5 _{D} = -3.5 s_{D}^{2} = 21
- Determine the values of t for which the null hypothesis, µ_{1} – µ_{2} = 0,
would be rejected in favor of the alternative hypotheses, µ_{1} – µ_{2}
< 0. Use α = .10.
- Conduct the paired difference test described in part a. Draw the appropriate conclusions.
- What assumptions are necessary so that the paired difference test will be valid?
- Find a 90% confidence interval for the mean difference µ_{D}.
- Which of the two inferential procedures, the confidence interval of part d of the test of hypothesis of part b, provides more information about the differences between the population means?
- Facility layout and material flow path design are major factors in the productivity analysis of automated manufacturing systems. Facility layout is concerned with the location arrangement o machines and buffers for work-in-process. Flow path design is concerned with the location arrangement of machines and buffers for work -in-process. Flow path design is concerned with the direction of manufacturing material flows (e.g., unidirectional or bidirectional). A manufacturer of printed circuit boards (PCBs) is interested in evaluating two alternative existing layout and flow path designs. The output of each design was monitored for eight consecutive working days.
Working Days | Design 1 | Design 2 |
8/16 | 1,220 units | 1,273 units |
8/17 | 1,092 units | 1,363 units |
8/18 | 1,136 units | 1.342 units |
8/19 | 1,205 units | 1,471 units |
8/20 | 1,086 units | 1,299 units |
8/23 | 1,274 units | 1,457 units |
8/24 | 1,145 units | 1,263 units |
8/25 | 1,281 units | 1,368 units |
- Construct a 95% confidence interval for the
difference in mean daily output of the two designs.
- What assumptions must hold to ensure the validity of the confidence interval?
- Design 2 appears to be superior to Design 1. Is this confirmed by the confidence interval?
- Construct a 95% confidence interval for (p_{1} – p_{2}) in each of the following situations:
- n_{1 }= 400, = 0.65; n_{2}= 400, = 0.58
- n_{1 }= 180, = 0.31; n_{2}= 250, = 0.25
- n_{1 }= 100, = 0.46; n_{2}= 120, = 0.61
- Suppose you want to estimate the difference between two population means correct to within 1.8 with a 95% confidence interval. If prior information suggests that the population variances are approximately equal to Ϭ_{1}^{2} = Ϭ_{2}^{2} = 14 and you want to select independent random samples of equal size from the population, how large should the sample sizes, n_{1} and n_{2}be?
- Determine each of the following F values:
- F_{.05} where v_{1}-= 9 and v_{2} = 6
- F_{.01} where v_{1}-= 18 and v_{2} = 14
- F_{.025} where v_{1}-= 11 and v_{2} = 4
- F_{.10} where v_{1}-= 20 and v_{2} = 5
- When new instruments are developed to perform chemical analyses of products (food, medicine etc.), they are usually evaluated with respect to two criteria: accuracy and precision. Accuracy refers to the ability of an instrument to identify correctly the nature and amounts of a product’s components. Precision refers to the consistency with which the instrument will identify the components of the same material. Thus, a large variability in the identification of a single batch of a product indicates a lack of precision. Suppose a pharmaceutical firm is considering two brands of an instrument designed to identify the components of certain drugs. As part of a comparison of precision, 10 test-tube samples of a well-mixed batch of a drug are selected and then five are analyzed by instrument A and five by instrument B. The data shown below are the percentages of the primary component of the drug given by the instruments. Do these data provide evidence of a difference in the precision of the two machines? Use α = 0.10.
Instrument A | Instrument B |
43 | 46 |
48 | 49 |
37 | 43 |
52 | 41 |
45 | 48 |