# problem solving exercise

This is a problem solving exercise that a certain major software company likes to use in their interviews. Like many algorithm design problems, there are many possible answers, but I have presented one of the more generally accepted ones.

A stream of integers between 1 and 1 million is received by a computer. The numbers arrive in random order, however each number only appears once in the sequence. How would you determine which one number was missing from the sequence?

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There are many ways to solve this problem, but the generally accepted one is to realize that the incoming integers comprise an arithmetic ...

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A problem solving exercise is practiced.

Various First Year Statistic Problems

(See attached file for full problem description)

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For Exercise 1, assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level.

1. N = 1200, x = 400, 99% confidence

For exercise 2, use the sample data and confidence level to construct the confidence interval estimate of the population proportion p.

2. N = 1200, x = 200, 99% confidence

For exercises 3a and b, use the given data to find the minimum sample size required to estimate a population proportion or percentage. Note: ^ should go "on top" of the p and q - I didn't know how to do it.

3. a. Margin of error: 0.038; confidence level: 95%; p^ and q^ unknown

b. Margin of error: three percentage points; confidence level: 90%; from a prior study, p^ is estimated by the decimal equivalent of 8%

For exercise 4, use the given confidence level and sample data to find (a) the margin of error E and (b) a confidence interval for estimating the population mean μ

4. Starting salaries of college graduates who have taken a statistics course: 95% confidence; n = 28, = $45,678, the population is normally distributed, and σ is known to be $9900

For exercise 5, use the given margin of error; confidence level, and population standard deviation σ to find the minimum sample size required to estimate an unknown population mean μ.

5. Margin of error: 3 lb, confidence level: 99%, σ = 15 lb

For exercise 6a and b, use this 95% confidence interval: (262.09, 374.11). The confidence interval results from using a sample of 80 measured cholesterol levels of randomly selected adults.

6. a. Express the confidence interval in the format of - E < μ < + E.

b. Write a statement that interprets the 95% confidence interval.

7. In order to help identify baby growth patterns that are unusual, we need to construct a confidence interval estimate of the mean head circumference of all babies that are two months old. A random sample of 100 babies is obtained, and the mean head circumference is found to be 40.6 cm. Assuming that the population standard deviation is known to be 1.6 cm, find a 99% confidence interval estimate of the mean head circumference of all two-month-old babies. What aspect of this problem is not realistic?

For problem 8a and b, does one of the following, as appropriate: (a) find the critical value Z /2, (b) find the critical value of t /2, (c) state that neither the normal nor the t distribution applies.

8. a. 90%; n = 9, σ = 4.2; population appears to be very skewed

b. 98%; n = 37; σ is unknown; population appears to be normally distributed

For exercise 9, construct the confidence interval.

9. A study was conducted to estimate hospital costs for accident victims who wore seat belts. Twenty randomly selected cases have a distribution that appears to be bell-shaped with a mean of $9004 and a standard deviation of $5629

a. Construct the 99% confidence interval for the mean of all such costs.

b. If you are a manager for an insurance company that provides lower rates for drivers who wear seat belts, and you want a conservative estimate for a worst case scenario, what amount should you use as the possible hospital cost for an accident victim who wears seat belts?

For problem 10a and b, find the critical z values. In each case, assume that the normal distribution applies.

10. a. = 0.10; H1 is p>0.18

b. = 0.005; H1 is p≠0.20

11. The claim is that the proportion of peas with yellow pods is equal to 0.25 (25%), and the sample statistics include n = 580 peas with 26.2% of them having yellow pods.

For problem 12a and b, state the final conclusion in simple nontechnical terms. Be sure to address the original claim.

12. a. Original claim: The proportion of college graduates who smoke is less than 0.27.

Initial conclusion: Reject the null hypothesis

b. Original claim: The proportion of M&Ms that are blue is equal to 0.10

Initial conclusion: Reject the null hypothesis

For exercises 13a, b, c, d, and e, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that address the original claim. Use the P-value method.

13. a. In a recent year, of the 109,857 arrests for Federal offenses, 29.1% were drug offenses. Use a 0.01 significance level to test the claim that that the drug offense rate is equal to 30%. How can the result be explained, given that 29.1% appears to be so close to 30%?

b. In 1990, 5.8% of job applicants who were tested for drugs failed the test. At the 0.01 significance level, test the claim that the failure rate is now lower if a simple random sample of 1520 current job applicants results in 58 failures. Does the result suggest that fewer job applicants now use drugs?

c. In one study of smokers who tried to quit smoking with nicotine patch therapy, 39 were smoking one year after the treatment, and 32 were not smoking one year after the treatment. Use a 0.10 significance level test to claim that among smokers who try to quit with nicotine patch therapy, the majority are smoking a year after the treatment. Do these results suggest that the nicotine patch therapy is ineffective?

d. The health of the bear population in Yellowstone National Park is monitored by periodic measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9 lb. Assuming that σ is known to be 121.8 lb, use a 0.10 significance level to test the claim that the population mean of all such bear weights is less than 200 lb.

e. . A random sample of 100 babies is obtained, and the mean head circumference is found to be 40.6 cm. Assuming that the population standard deviation is known to be 1.6 cm, use a 0.05 significance level to test the claim that the mean head circumference of all two-month-old babies is equal to 40.0 cm.

For exercise 14a and b, find the test statistic, P-value, critical value(s), and state the final conclusion.

14. a. Claim: The mean starting salary for college graduates who have taken a statistics course is equal to $46,000. Sample data: n = 65, = $45,678. Assume that σ = $9900 and the significance level is = 0.05

b. Claim: The mean starting salary for college graduates who have taken a statistics course is equal to $46,000. Sample data: n = 27, = $45,678, s = $9900. The significance level is = 0.05

For exercise 15a and b, determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither.

15. a. claim: μ = 75. Sample data: n = 25, = 102, s = 15.3. The sample data appears to come from a population with a distribution that is very far from normal, and σ is unknown.

b. Claim: μ = 2.80. Sample data: n = 150, = 2.88, s = 0.24. The sample data appears to come from a population with a distribution that is not normal, and σ is unknown.

For exercise 16, assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use either the traditional value or P-value method for testing hypotheses.

16. Sugar content for a sample of different cereals is summarized with these statistics: n = 16, = 0.295g, s = 0.168g. Use a 0.05 significance level to test the claim of a cereal lobbyist that the mean for all cereals is less than 0.3g.

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(See attached file for full problem description)

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