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# Confidence interval and 6 degrees of freedom

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#1.
8.15:
Using Table A.4 (page 640), find t.100, t.025, and t.001 based on 11 degrees of freedom. Also, find these t points based on 6 degrees of freedom.

#2.
8.16:
Suppose that for a sample of n = 11 measurements, we find that _ = 72 and s 5. Assuming normality, compute confidence intervals for the population mean ? with the following levels of confidence: x¯

a: 95%
b: 99%
c: 80%
d: 90%
e: 98%
f: 99.8%

#3.
8.17:
The bad debt ratio for a financial institution is defined to be the dollar value of loans defaulted divided by the total dollar value of all loans made. Suppose a random sample of seven Ohio banks is selected and that the bad debt ratios (written as percentages) for these banks are 7 percent, 4 percent, 6 percent, 7 percent, 5 percent, 4 percent, and 9 percent. Assuming the bad debt ratios are approximately normally distributed, the MINITAB output of a 95 percent confidence interval for the mean bad debt ratio of all Ohio banks is as follows:

Variance: D-Ratio
N: 7
Mean: 6.00000
St. Dev: 1.82574
SE Mean: 0.69007
95% CI: (4.31147, 7.68853)

A: Using the Xbar and s on the MINITAB output, verify the calculation of the 95 percent confidence interval, and calculate a 99 percent confidence interval for the mean debt-to-equity ratio.

#4.
8.19
Air traffic controllers have the crucial task of ensuring that aircraft don't collide. To do this, they must quickly discern when two planes are about to enter the same air space at the same time. They are aided by video display panels that track the aircraft in their sector and alert the controller when two flight paths are about to converge. The display panel currently in use has a mean "alert time" of 15 seconds. (The alert time is the time elapsing between the instant when two aircraft enter into a collision course and when a controller initiates a call to reroute the planes.) According to Ralph Rudd, a supervisor of air traffic controllers at the Greater Cincinnati International Airport, a new display panel has been developed that uses artificial intelligence to project a plane's current flight path into the future. This new panel provides air traffic controllers with an earlier warning that a collision is likely. It is hoped that the mean "alert time," ?, for the new panel is less than 8 seconds. In order to test the new panel, 15 randomly selected air traffic controllers are trained to use the panel and their alert times for a simulated collision course are recorded. The sample alert times (in seconds) are: 7.2, 7.5, 8.0, 6.8, 7.2, 8.4, 5.3, 7.3, 7.6, 7.1, 9.4, 6.4, 7.9, 6.2, 8.7.

A: Using the fact that _ = 7.4 and s = 1.026, find a 95 percent confidence interval for the population mean alert time, ?, for the new panel.

#5.
8.21: A production supervisor at a major chemical company wishes to determine whether a new catalyst, catalyst XA-100, increases the mean hourly yield of a chemical process beyond the current mean hourly yield, which is known to be roughly equal to, but no more than, 750 pounds per hour. To test the new catalyst, five trial runs using catalyst XA-100 are made. The resulting yields for the trial runs (in pounds per hour) are 801, 814, 784, 836, and 820. Assuming that all factors affecting yields of the process have been held as constant as possible during the test runs, it is reasonable to regard the five yields obtained using the new catalyst as a random sample from the population of all possible yields that would be obtained by using the new catalyst. Furthermore, we will assume that this population is approximately normally distributed.

A: Using the Excel output in Figure 8.13, find a 95 percent confidence interval for the mean of all possible yields obtained using catalyst XA-100.

https://brainmass.com/statistics/confidence-interval/confidence-interval-6-degrees-freedom-449371

#### Solution Summary

Answers of multiple choice questions on confidence interval and control charts

\$2.19

## Statistics: probability, confidence interval, mean, standard deviation, test statistic

See attached four problems.

1. Avery short quiz has one multiple choice questions with five possible choices (a, b, c, d, e) and one true or false question. Assume you are taking the quiz but do not have any idea what the correct answer is to either question, but you mark an answer any way.

a. What is the probability that you have given the correct answer to both questions?
b. What is the probability that only one of the two answers is correct?
c. What is the probability that neither answer is correct?
d. What is the probability that only your answer to the multiple choice question is correct?
e. What is the probability that you have only answered the true or false question correctly?

2. Information regarding the price of a roll of camera film (35 mm, 24 exposure) for a sample of 12 cities world wide is shown below. Determine 91% confidence interval for the population mean.

Price of film information is given in the attachment.

3. Confirmed cases of West Nile virus in birds for a sample of six countries in the state of Georgia are shown below.

Country Cases

Catoosa 6
Chattoogan 3
Gordon 5
Murray 3
Walker 4

You want to determine if the average number of cases of West Nile virus in the state of Georgia is significantly more than 3. Assume the population is normally distributed.

a. State the null and alternative hypotheses
b. Compute the mean and the standard deviation of the sample.
c. Compute the standard error of the mean.
d. Determine the test statistic.
e. Determine the P-value and 95% confidence, test the hypotheses.

4. Consider the following hypotheses test.

H0: mu >= 80
Ha: mu < 80

A sample of 121 provided a sample mean of 77.3. The population statndard deviation is known to be 16.5.

a. Compute the value of the test statistic.
b. Determine the P-value; and at 93.7% confidence, test the above hypotheses.
c. Using the critical value approach at 93.7% confidence, test the hypotheses.

5. In the last Presidential election, a national survey company claimed that no more than 50% (i.e., <=50%) of all registered voters voted for the Republican candidate. In a random sample of 400 registered voters, 208 voted for the Republican candidate.

a. State the null and alternative hypotheses
b. Compute the test statistic.
c. At 95% confidence, compute the P-value and test the hypotheses.

6. Consider the following hypothesis test:

H0: mu <= 38
Ha: mu > 38

You are given the following information obtained from a random sample of six observations. Assume the population has a normal distribution.

X
38
40
42
32
46
42
a. Compute the mean of the sample.
b. Determine the standard deviation of the sample.
c. Determine the standard error of the mean.
d. Compute the value of the test statistic.
e. At 95% confidence using the P-value approach, test the above hypotheses.

7. A test on world history was given to a group of individuals before and also after a film on the history of the world was presented. The results are given in the attachment. We want to determine if the film significantly increased the test scores.

a. Give the hypotheses for this problem.
b. Compute the test statistic.
c. At 95% confidence, test the hypotheses.

8. The Dean of students at UTC has said that the average grade of UTC students is higher than that of the students at GSU. Random samples of grades from the two schools are selected, and the results are shown in the attachment.

a. Give the hypotheses.
b. Compute the degrees of freedom for this test.
c. Compute the test statistic.
d. At a 0.1 level of significance, test the Dean of Student's statement.

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