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Chapter 5: Q4E (page 303)
Refer to Exercise 5.3 and find . Then use the sampling distribution offound in Exercise 5.3 to find the expected value of. Note that.
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Question:Fingerprint expertise. Refer to the Psychological Science (August 2011) study of fingerprint identification, Exercise 4.53 (p. 239). Recall that when presented with prints from the same individual, a fingerprint expert will correctly identify the match 92% of the time. Consider a forensic database of 1,000 different pairs of fingerprints, where each pair is a match.
a. What proportion of the 1,000 pairs would you expect an expert to correctly identify as a match?
b. What is the probability that an expert will correctly identify fewer than 900 of the fingerprint matches?
Question:A random sample of 40 observations is to be drawn from a large population of measurements. It is known that 30% of the measurements in the population are 1s, 20% are 2s, 20% are 3s, and 30% are 4s.
a. Give the mean and standard deviation of the (repeated) sampling distribution of, the sample mean of the 40 observations.
b. Describe the shape of the sampling distribution of. Does youranswer depend on the sample size?
Variable life insurance return rates. Refer to the International Journal of Statistical Distributions (Vol. 1, 2015) study of a variable life insurance policy, Exercise 4.97 (p. 262). Recall that a ratio (x) of the rates of return on the investment for two consecutive years was shown to have a normal distribution, with , . Consider a random sample of 100 variable life insurance policies and letrepresent the mean ratio for the sample.
a. Find E(x) and interpret its value.
b. Find Var(x).
c. Describe the shape of the sampling distribution of.
d. Find the z-score for the value .
e. Find
f. Would your answers to parts a鈥揺 change if the rates (x) of return on the investment for two consecutive years was not normally distributed? Explain.
Fecal pollution at Huntington Beach. California mandates fecal indicator bacteria monitoring at all public beaches. When the concentration of fecal bacteria in the water exceeds a certain limit (400 colony-forming units of fecal coliform per 100 millilitres), local health officials must post a sign (called surf zone posting) warning beachgoers of potential health risks. For fecal bacteria, the state uses a single-sample standard; if the fecal limit is exceeded in a single sample of water, surf zone posting is mandatory. This single-sample standard policy has led to a recent rash of beach closures in California. A study of the surf water quality at Huntington Beach in California was published in Environmental Science & Technology (September 2004). The researchers found that beach closings were occurring despite low pollution levels in some instances, while in others, signs were not posted when the fecal limit was exceeded. They attributed these "surf zone posting errors" to the variable nature of water quality in the surf zone (for example, fecal bacteria concentration tends to be higher during ebb tide and at night) and the inherent time delay between when a water sample is collected and when a sign is posted or removed. To prevent posting errors, the researchers recommend using an averaging method rather than a single sample to determine unsafe water quality. (For example, one simple averaging method is to take a random sample of multiple water specimens and compare the average fecal bacteria level of the sample with the limit of 400 CFU/100 mL to determine whether the water is safe.) Discuss the pros and cons of using the single sample standard versus the averaging method. Part of your discussion should address the probability of posting a sign when the water is safe and the probability of posting a sign when the water is unsafe. (Assume that the fecal bacteria concentrations of water specimens at Huntington Beach follow an approximately normal distribution.
Use the computer to generate 500 samples, each containing n = 25 measurements, from a population that contains values of x equal to 1, 2, . . 48, 49, 50 Assume that these values of x are equally likely. Calculate the sample mean and median m for each sample. Construct relative frequency histograms for the 500 values of and the 500 values of m. Use these approximations to the sampling distributions of and m to answer the following questions:
a. Does it appear that and m are unbiased estimators of the population mean? [Note:]
b. Which sampling distribution displays greater variation?
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