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Chapter 5: Q13E (page 314)
Will the sampling distribution of always be approximately normally distributed? Explain
A sampling distribution is statistics derived by continuous sampling from a greater populace.
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Question: Refer to Exercise 5.5, in which we found the sampling distribution of the sample median. Is the median an unbiased estimator of the population mean m?
Suppose a random sample of n = 500 measurements is selected from a binomial population with probability of success p. For each of the following values of p, give the mean and standard deviation of the sampling distribution of the sample proportion,.
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.
Suppose a random sample of n = 25 measurements are selected from a population with mean and standard deviation s. For each of the following values of and role="math" localid="1651468116840" , give the values of and .
Critical-part failures in NASCAR vehicles. Refer to The Sport Journal (Winter 2007) analysis of critical-part failures at NASCAR races, Exercise 4.144 (p. 277). Recall that researchers found that the time x (in hours) until the first critical-part failure is exponentially distributed with = .10 and s = .10. Now consider a random sample of n = 50 NASCAR races and let represent the sample meantime until the first critical-part failure.
a) Find and
b) Although x has an exponential distribution, the sampling distribution of x is approximately normal. Why?
c) Find the probability that the sample meantime until the first critical-part failure exceeds .13 hour.
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