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Week 4 assignment is focused on inferences you can make
regarding a single sample, whether you have a “large” sample of more than 30
items or a “small” sample of 30 items or fewer. You will formulate a
confidence interval to estimate the population mean for both a large and small
sample as well as for a large sample population proportion. In addition,
you will construct a hypothesis test for a population mean, again for both a
large and small sample and a large sample population proportion. **** Please answere the question in the attached documents.
***Week 5’s assignment is quite similar to that of Week
4. Instead on inferences, you can make regarding a single
sample, you will make inferences regarding two samples. Once again,
you will formulate a confidence interval to estimate the population
means. You will consider two independent samples as well as two paired
samples, for example, a sample that represents some characteristic before and
after the introduction of some stimulus. You will also study two
population proportions.
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TIM7100
DATA FILE 4
Assignment 4
1. What is the confidence level for each of the following confidence intervals for µ?
a. 𝑥̅ ± 1.96 (𝛿⁄ )
√𝑛
𝛿
b. 𝑥̅ ± 1.645 ( ⁄ )
√𝑛
c. 𝑥̅ ± 2.575 (𝛿⁄ )
√𝑛
d. 𝑥̅ ± 1.282 (𝛿⁄ )
√𝑛
𝛿
e. 𝑥̅ ± 0.99 ( ⁄ )
√𝑛
2. A random sample of n measurements was selected from a population with unknown mean µ
and standard deviation Ϭ. Calculate a 95% confidence interval for µ for each of the following
situations:
a. n= 75, 𝑥̅ = 28, s2 = 12
b. n= 200, 𝑥̅ = 102, s2 = 22
c. n= 100, 𝑥̅ = 15, s = 3
d. n= 100, 𝑥̅ =4.05, s = 0.83
e. Is the assumption that the underlying population of measurements is normally
distributed necessary to ensure the validity of the confidence intervals in parts a -d?
Explain.
3. Named for the section of the 1978 Internal Revenue Code that authorized them, 401(k) plans
permit employees to shift part of their before-tax salaries into investments such as mutual
funds. One company, concerned with what it believed was a low employee participation rate in
its 401(k) plan, sampled 30 other companies with similar plans and asked for their 401(k)
participation rates. The following rates (in percentages) were obtained:
80
76
81
77
82
80
85
60
80
79
82
70
88
85
80
79
83
75
87
78
80
84
72
75
90
84
82
77
75
86
Descriptive statistics from SPSS for the data are as follows:
Number of Valid Observations (Listwise)
=
Variable
PARTRATE
Minimum
60.00
Mean
79.73
Std Dev
5.96
30
Maximum
90.00
N Label
30
a. Use the SPSS information above to construct a 95% confidence interval for the mean
participation rate for all companies that have 401(k) plans.
b. Interpret the interval in the context of this problem.
c. What assumption is necessary to ensure the validity of this confidence interval?
d. If the company that conducted the sample has a 71% participation rate, can it safely conclude
that its rate is below the population mean rate for all companies with 401(k) plans? Explain.
e. If in the data set the 60% had been 80%, how would the center and width of the confidence
interval you constructed in part a be affected?
4. Suppose you have selected a random sample of n = 5 measurements from a normal distribution.
Compare the standard normal z values with the corresponding t values if you were forming the
following confidence intervals:
a. 80% confidence interval
b. 90% confidence interval
c. 95% confidence interval
d. 98% confidence interval
e. 99% confidence interval
5. The following random sample was selected from a normal distribution: 4, 6, 3, 5, 9, 3.
a. Construct a 90% confidence interval for the population mean µ
b. Construct a 95% confidence interval for the population mean µ.
c. Construct a 99% confidence interval for the population mean µ
d. Assume that the sample mean 𝑥̅ and sample standard deviations remain exactly the
same as those you just calculated but that they are based on a sample of n = 25
observations rather than n = 6 observations. Repeat parts a – c. What I the effect of
increasing the sample size on the width of the confidence intervals?
Increasing the sample size decreases the width of the confidence interval.
6. The following is a 90% confidence interval for p: (0.26, 0.54). How large was the sample used to
construct this interval?
7. When companies employ control charts to monitor the quality of their products, a series of
small samples is typically used to determine if the process is “in control” during the period of
time in which each sample is selected. Suppose a concrete-block manufacturer samples nine
blocks per hour and tests the breaking strength of each. During one-hour’s test, the mean and
standard deviation are 985.6 pounds per square inch (psi) and 22.9 psi, respectively.
a. Construct a 99% confidence interval for the mean breaking strength of blocks produced
during the hour in which the sample was selected.
b. The process is to be considered “out of control” if the mean strength differs from 1,000
psi. What would you conclude based on the confidence interval constructed in part a?
c. Repeat parts a and b using a 90% confidence interval.
d. The manufacturer wants to be reasonably certain that the process is really out of
control before shutting down the process and trying to determine the problem. Which
interval, the 99% or 90% confidence interval, is more appropriate for making the
decision? Explain.
e. Which assumptions are necessary to ensure the validity of the confidence intervals?
8. Answer each of the following:
a. Which hypothesis, the null or the alternative, is the status-quo hypothesis?
Which is the research hypothesis?
b. Which element of a test of hypothesis is used to decide whether to reject the null
hypothesis in favor of the alternative hypothesis?
c. What is the level of significance called of a test of hypothesis?
d. What is the difference between Type I and Type II errors in hypothesis testing? How do
a and b relate to Type I and Type II errors?
e. List the four possible results of the combinations of decisions and true states of nature
for a test of hypothesis?
9. A random sample of 100 observations from a population with standard deviation 60 yielded a
sample mean of 110.
a. Test the null hypothesis that µ = 100 against the alternative hypothesis that µ > 100
using α = 0.05. Interpret the results of the test.
b. Test the null hypothesis that µ = 100 against the alternative hypothesis that µ ≠ 100
using α = 0.05. Interpret the results of the test.
c. Compare the results of the two tests you conducted. Explain why the results differ.
10. A company has devised a new toner cartridge for its laser jet home/office printer that it believes
has a longer lifetime (on average) than the one currently being produced. To investigate its
length of life, 225 of the new cartridges were tested by counting the number of high-quality
printed pages each was able to produce. The sample mean and standard deviation were
determined to be 1,511.4 pages and 35.7 pages, respectively. The historical average lifetime for
cartridges produced by the current process is 1,502.5 pages; the historical standard deviation is
97.3 pages.
a. What are the appropriate null and alternative hypotheses to test whether the mean
lifetime of the new cartridges exceeds that of the old cartridges?
b. Use α = 0.05 to conduct the test in part a. Do the new cartridges have an average
lifetime that is statistically significantly longer than the cartridges currently in
production?
c. Does the difference in average lifetimes appear to be of practical significance from the
perspective of the consumer? Explain.
d. Should the apparent decrease in the standard deviation in lifetimes associated with the
new cartridges be viewed as an improvement over the old cartridges?
11. For each α and observed significance level (p-value) pair, indicate whether the null hypothesis
would be rejected.
a. α= .05, p-value = .10
b. α= .10, p-value = .05
c. α= .01, p-value = .001
d. α= .025, p-value = .05
e. α= .10, p-value = .45
12. A random sample of 50 consumers taste tested a new snack food. Their responses were coded
(0: do not like; 1: like, 2: indifferent) and recorded below:
a. Test H0: p = 0.5 against Ha: p > 0.5, where p is the proportion of customers who do not
like the snack food. Use α = 0.10.
b. Find the observed significance level of your test.
1
0
0
1
2
0
1
1
0
0
0
1
0
2
0
2
2
0
0
1
1
0
0
0
0
1
0
2
0
0
0
1
0
0
1
0
0
1
0
1
0
2
0
0
1
1
0
0
0
1
TIM7100
DATA FILE 5
ASSIGNMENT 5
1. Independent random samples of 64 observations each are chosen from two normal
populations with the following means and standard deviations:
Population 1
Population 2
µ1 = 12
µ2 = 10
Ϭ1 = 4
Ϭ2 = 3
Let 𝑥̅ 1 and 𝑥̅ 2 denote the two sample means.
a. Give the mean and standard deviation of the sampling distribution of 𝑥̅ 1.
b. Give the mean and standard deviation of the sampling distribution of 𝑥̅ 2.
c. Suppose you were to calculate the difference (𝑥̅ 1 – 𝑥̅ 2) between the sample means.
Find the mean and standard deviation of the sampling distribution of (𝑥̅ 1 – 𝑥̅ 2).
d. Will the statistic (𝑥̅ 1 – 𝑥̅ 2) be normally distributed? Explain.
2. In order to compare the means of two populations, independent random samples of
400 observations are selected from each population, with the following results:
a.
b.
c.
d.
e.
Sample 1
Sample 2
𝑥̅ 1= 5,275
𝑥̅ 2 = 5,240
s1 = 150
s2 = 200
Use a 95% confidence interval to estimate the difference between the population
means (µ1 – µ2). Interpret the confidence interval.
Test the null hypothesis H0: (µ1 – µ2) = 0 versus the alternative hypothesis Ha: (µ1 µ2) ≠ 0. Give the significance level of the test, and interpret the result.
Suppose the test in part b was conducted with the alternative hypothesis Ha: (µ1 µ2) > 0. How would your answer to part b change?
Test the null hypothesis H0: (µ1 – µ2) = 25 versus the alternative hypothesis Ha: (µ1 µ2) ≠ 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?.
3. Assume that Ϭ12 = Ϭ22 = Ϭ2. Calculate the pooled estimator of Ϭ2 for each of the
following cases:
a. s12 = 120, s22 = 100, n1 = n2 = 25
b. s12 = 12, s22 = 20, n1 =20, n2 = 10
c. s12 = 0.15, s22 = 0.20, n1 =6, n2 = 10
d. s12 = 3,000, s22 =2,500, n1 =16, n2 = 17
e. 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?
4. 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.
30.1
28.2
Plant Discharge
36.2
29.8
33.4
34.9
29.7
27.3
Upstream
30.3
31.7
26.4
32.3
a. Why might the bacteria counts shown here tend to be approximately normally
distributed?
b. 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.
c. What assumptions are necessary to ensure the validity of this test?
5. A paired difference experiment produced the following data:
nD = 18
𝑥̅ 1 = 92
𝑥̅ 2 = 95.5
𝑥̅ D = -3.5
sD2 = 21
a. Determine the values of t for which the null hypothesis, µ1 – µ2 = 0, would be
rejected in favor of the alternative hypotheses, µ1 – µ2 < 0. Use α = .10. b. Conduct the paired difference test described in part a. Draw the appropriate conclusions. c. What assumptions are necessary so that the paired difference test will be valid? d. Find a 90% confidence interval for the mean difference µD. e. 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? 6. 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 -inprocess. 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 8/16 8/17 8/18 8/19 8/20 8/23 8/24 8/25 Design 1 1,220 units 1,092 units 1,136 units 1,205 units 1,086 units 1,274 units 1,145 units 1,281 units Design 2 1,273 units 1,363 units 1.342 units 1,471 units 1,299 units 1,457 units 1,263 units 1,368 units a. Construct a 95% confidence interval for the difference in mean daily output of the two designs. b. What assumptions must hold to ensure the validity of the confidence interval? c. Design 2 appears to be superior to Design 1. Is this confirmed by the confidence interval? 7. Construct a 95% confidence interval for (p1 – p2) in each of the following situations: a. n1 = 400, 𝑝̂1= 0.65; n2= 400, 𝑝̂2 = 0.58 b. n1 = 180, 𝑝̂1= 0.31; n2= 250, 𝑝̂2 = 0.25 c. n1 = 100, 𝑝̂1= 0.46; n2= 120, 𝑝̂2 = 0.61 8. 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 Ϭ12 = Ϭ22 = 14 and you want to select independent random samples of equal size from the population, how large should the sample sizes, n1 and n2 be? 9. Determine each of the following F values: a. F.05 where v1 -= 9 and v2 = 6 b. F.01 where v1 -= 18 and v2 = 14 c. F.025 where v1 -= 11 and v2 = 4 d. F.10 where v1 -= 20 and v2 = 5 10. 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 43 48 37 52 45 Instrument B 46 49 43 41 48 ... Purchase answer to see full attachment