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Standard Error, Confidence Interval and Hypothesis Testing

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I need some help with these hypothesis testing and standard error questions:
1. The mean of a normal probability distribution is 60; the standard deviation is 5.
(a) About what percent of the observations lie between 55 and 65?
(b) About what percent of the observations lie between 50 and 70?
(c) About what percent of the observations lie between 45 and 75?

2. A normal population has a mean of 12.2 and a standard deviation of 2.5.
1. Compute the z value associated with 14.3.
2. What proportion of the population is between 12.2 and 14.3?
3. What proportion of the population is less than 10.0?

3. The amounts of money requested on home loan applications at Down River Federal Savings follow the normal distribution, with a mean of $70,000 and a standard deviation of $20,000. A loan application is received this morning. What is the probability:
1. The amount requested is $80,000 or more?
2. 2. The amount requested is between $65,000 and $80,000?
3. The amount requested is $65,000 or more?

4. Jon Molnar will graduate from Carolina Forest High School this year. He took the American College Test (ACT) for college admission and received a score of 30. The high school principal informed him that only 2 percent of the students taking the exam receive a higher score. The mean score for all students taking the exam is 18.3. Jon's friends Karrie and George also took the test but were not given any information by the principal other than their scores. Karrie scored 25 and George 18. On the basis of this information, what were Karrie's and George's percentile ranks? Assume that the distribution of scores follows the normal distribution.

5. A population consists of the following five values: 2, 2, 4, 4, and 8.
1. List all samples of size 2, and compute the mean of each sample.
2. Compute the mean of the distribution of sample means and the population mean. Compare the two values.
3. Compare the dispersion in the population with that of the sample means.

6. A processor of carrots cuts the green top off each carrot, washes the carrots, and inserts six to a package. Twenty packages are inserted in a box for shipment. To test the weight of the boxes, a few were checked. The mean weight was 20.4 pounds, the standard deviation 0.5 pounds. How many boxes must the processor sample to be 95 percent confident that the sample mean does not differ from the population mean by more than 0.2 pounds?

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This solution contains detailed explanations and calculations to answer the six statistical problem sets.

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Solving hypothesis testing and confidence interval problems

Exercise 8.5
Find a confidence interval for u assuming that each sample is from a normal population.
a. x = 14, sigma = 4, n = 5, 90 percent confidence
b. x = 37, sigma = 15, 99 percent confidence
c. x = 121, sigma = 15, 95 percent confidence

Exercise 9.19
Calculate the test statistic and p-value for each sample.
a. H0 = u = 60 versus H1: u does not equal 60, sigma = 0.025, x = 63, sigma = 8, n = 16
b. H0: u is greater than or equal to 60 versus H1: u < 60, sigma = 0.05, x = 58, sigma = 5, n = 25
c. H0: u is less than or equal to H1: u > 60, sigma = 0.05, x = 65, sigma = 8, n =36

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