Problem 3.1:

We've established that heights of 12-year-old boys vary according to a normal distribution with mu =145 cm and sigma = 8 cm. (try using both the tables and Stata- and drawing pictures is helpful, though there is no need to submit them. This problem is adapted from Gerstman 7.5, so if you need to figure out how to approach it, look at that problem and the solutions in the textbook)

a) What proportion of this population is less than 155 cm tall?

b) What proportion is less than 140 cm in height?

c) What proportion is between 140 and 155 cm?

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Problem 3.2

a) What is the 40^th percentile on a Standard Normal Distribution?b) What is the 40^th percentile in the Normal distribution of heights described in Problem 3.1?

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Problem 3.3

Set up null and alternative hypotheses for each of these claims. Use two-sided alternative hypotheses in each instance. (Note: while not required, it is good practice to write the hypotheses in both symbols AND words. This problem is derived from Gerstman 9.2).

a) The average single-family home price in Orange County, NC was $340,000 in 2019. A real estate agent claims that price has increased since that time.b) Students who watch the 711 recorded lectures at original speed score an average of 88% on the exams. Students who slow down the lectures claim that their exam scores are higher.c) A typical gas grill takes 12 minutes to preheat. A new grill with a "flash preheat" setting claims to reduce that time.

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Problem 3.4

About 22% of a population takes multivitamins. Our study has 10,000 participants from this population. X is the number in the sample who take multivitamins. In our study we compute p-hat= = proportion in sample who take multivitamins.

a) What is the distribution of X? (Hint: Think back to the previous chapter.)

b) What is the approximate distribution of p-hat? (Shape, mean, s.d....)

c) Using the 68-95-99.7 rule, state an interval that we would expect to phat to fall between about 95% of the time.

d) If you found that phat = 0.24, would you be surprised with how large this value is? (Compute the P(phat>0.24) and comment on whether the probability is "small "- which means it is an unusual event). This is foreshadowing of p-values.

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Problem 3.5

In 2021, SAT scores were approximately normally distributed with a mean of 1500 and a standard deviation of 300. Consider the population of all students who took the SAT in 2021.

a) What is the probability a random person sampled from thispopulation has a SAT score that is greater than 1700?

b) You take a simple random sample of n=25 individuals from this population and calculate the mean SAT score of the sample. Describe the sampling distribution of x-bar.

c) In the scenario described in b), what is the probability of getting a sample mean that is greater than 1700? [Determine Pr(x-bar >1700).] d) Which is more unlikely: observing a single value above 1700 or observing an x-bar above 1700 from a sample of n=25. Why?

e) State the values of the standard deviation and the standard error for the situation described above. Explain in simple English, what is the difference between the standard deviation and the standard error in the context of this problem.

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Problem 3.6 (adapted from Gerstman 9.3, p. 202)

In 2021, SAT scores were approximately normally distributed with a mean of 1500 and a standard deviation of 300. Suppose you obtain a random sample of n=25 people who took the SATin 2022.

a) We seek evidence against the hypothesis that the 2022 sample comes from a population with =1500. Sketch the curve that describes the sampling distribution of x-bar under the null hypothesis. Mark the horizontal axis with values that are 1, 2, and 3 standard errors above and below the mean. (You can assume that sigma is still 300).

b) Suppose a sample finds an x-bar of 1437. Mark this finding on the horizontal axis of your sketch. Then calculate the z-value for the result. Does this observation provide strong evidence against H₀?

c) Building on your work from parts a and b, write this scenario as a hypothesis test using the steps we discussed in class. Suppose we are wondering whether scores are lower in 2022 than 2021; use a one-sided alternative hypothesis. Fill in the following table to make sure you get all 5 steps:

Hypotheses
Data
Test Statistic (z-value)
p-value and interpretation
Conclusion

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Problem 3.7 (Gerstman 8.3, p. 178)

Say whether each of the underlined numbers is a parameter or a statistic.

a) A data set based on 168 hospital discharge summaries shows that 20% of patients were uninsured. (The review takes place in a large referral hospital.)

b) Data from the complete enumeration of a standard metropolitan area (SMA) indicate that 12% of the inhabitants are African American. A telephone survey based on random-digit-dialing of individuals from this SMA found that 8% of the respondents were African American.

c) Data from 10 online pharmacies reveal that the average cost of a 1-month supply of a particular medication is $31.20 with standard deviation $7.75. Data from 10 community pharmacies shows an average cost of $33.18 with standard deviation $7.88

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Problem 3.8 (Adapted from Gerstman 8.8)

Suppose you could take all possible samples of size n = 16 from a normal population with = 50 and = 4. (Note: You may use Stata or a Normal table if you like, but this problem can be completed without them.)

a) Sketch or describe in words the sampling distribution of x-bar. Identify values on the horizontal axis of this sampling distribution that are 1 and 2 standard deviations around its center.

b) Would you be surprised to find a sample mean of 47 under these conditions? Explain your reasoning.

c) Would you be surprised to find a sample mean of 51.5? Explain your reasoning.

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Problem 3.9 (Gerstman 8.7, p. 185)

A laboratory kit states that the standard deviation of its results can be expected to vary with =1 unit. A lab technician takes four measurements using this kit.

a) What is the value of the standard deviation of the x-bar of the four measurements taken with this kit?

b) Explain to a lay person the advantage of reporting the average of four measurements rather than using a single measurement.

c) How many times must we repeat the measurement before we obtain a standard deviation of the mean measurement of 0.2 units? [Hint: Rearrange the SE, formula to solve for n.]

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Problem 3.10 (Gerstman 9.10, p. 208)

Lithium carbonate is a drug used to treat bipolar mental disorders. The average dose in well-maintained patients is 1.3 mEq/L with standard deviation 0.3 mEq/L. A random sample of 25 patients on lithium demonstrates a mean level (x-bar) of 1.4 mEq/L. Test to see whether this mean is significantly higher than that of a well-maintained patient population. Use a two-sided alternative hypothesis for your test. Show all hypothesis testing steps. Comment on your findings.