Exam 9: Estimation Using a Single Sample

The choice of where to build nests is an important one for small bats. Their survival and reproduction depends on making good choices. Research has suggested that the warmth of the nest location is important, as well as protection from predators. In a study of bats nesting under bridges over one and two lane roads in Louisiana, researchers investigated the theory that nest location is affected by danger from predators. The researchers reasoned that if bats were seeking to avoid predators they would roost in the narrowest spaces available under the bridge. a) Forty-seven nests of a particular species were located under bridges on one-lane roads. Of these 47 nests, 28 of them were under the narrowest spaces of the bridges. Determine whether or not the sample size is large enough to use the standard normal distribution to find the margin of error. Be sure to show any supporting calculations. When this species of bat nested under bridges on two-lane roads, 24 out of 38 nests were found in the narrowest spaces. b) Assuming these nests to be representative of nests of this species under two-lane bridges, estimate the population proportion of nests that are in the narrowest of spaces. c) On the basis of prior observations and theory, the researchers believed that the proportion of nests located in the narrowest spaces is $p = 0.30$ . If their belief is correct, what is the standard error of $\hat { p }$ ? d) Calculate a $95 \%$ confidence interval for $p$ , the proportion of the nests of this species choosing the narrowest spaces when nesting under two-lane bridges. e) Suppose the researchers wished to estimate the proportion of nests of this species that would be in the narrowest spaces of bridges when under two-lane roads. What sample size would be needed to obtain a margin of error of $0.05$ with $95 \%$ confidence?

The large sample confidence interval formula for estimating p can safely be used whenever $n \geq 30$

What does the confidence level tell you about a confidence interval?

For small samples, the margin of error is less than the standard error of $\hat { p }$

When conducting research into the efficacy of drugs to combat psychological disorders it is very common to use double blinding as a design strategy. In a double blind study neither the patient nor the doctor knows whether the patient is receiving the treatment or the placebo drug. Does double blinding actually work? That is, is the doctor completely ignorant of the patient's assignment to their treatment group? a) In a study involving the use of the drugs alprazolam and propranolol, 43 patients with panic attack disorder and agoraphobia were treated. After completing the study the physicians involved were asked to guess the treatments. The physicians correctly guessed the treatments for 30 patients. Determine whether or not the sample size is large enough to use the standard normal distribution to find the margin of error. Be sure to show any supporting calculations. In a different study of double-blinding, the use of lithium as a treatment to counteract aggressive tendencies in adolescents was under study. In that study the physicians correctly guessed the treatment in 37 out of 57 patients. b) Assuming these adolescents and doctors to be representative, estimate the population proportion of patients for whom the doctor could correctly guess the treatment. c) If the physicians were simply guessing about the treatments administered to the patients, $p = 0.50$ would be the proportion of correct guesses. If the physicians were guessing, what would be the standard error of $\hat { p }$ ? d) Based on the given data for the lithium treatment, calculate a $95 \%$ confidence interval for $p$ , the proportion of correct guesses by physicians. e) Suppose the researchers wished to get a more precise estimate of the proportion of correct guesses of treatment when using lithium in this context. What sample size would be needed to obtain a margin of error of $0.05$ with $95 \%$ confidence?

$\hat { p }$ is a biased estimator.

Eyestrain is thought to be associated with different types of office work. The Acme Temp-Help Company provides short-term employees to substitute for vacationing data entry workers. Previous research has shown that approximately 21% of full time data entry workers have eyestrain, and Acme would like to estimate the proportion, p, of their 6,000 part-time data entry employees that have eyestrain. A) They would like to estimate p to within 0.05 with 95% confidence. If they accept the value of 0.21 for full-time data entry workers as a reasonable initial estimate of p, what sample size should they use for their study? B) Suppose Acme believed that the working conditions of their temporary data entry personnel are different enough that they should not depend on the 0.21 as an initial estimate. In a few sentences, explain how your procedure for choosing a samples size would differ from your solution in part (a). (Do NOT recalculate a new estimate of the necessary sample size!)

In general, a wider confidence interval is associated with greater precision.

In a few sentences, identify the two desirable characteristics of a good estimator, and explain what makes them desirable characteristics.

$\hat { p }$ is an estimator.

Briefly define the following terms in your own words: A) Confidence interval B) The confidence level associated with a confidence interval C) The margin of error associated with a confidence interval

An understanding of choices of habitat is an important element in understanding the biology and ecology of birds. One aspect of habitat choice is the pattern of preferred vertical height when feeding in forests. In a study of small birds in an old-growth forest in the Pacific Northwest, researchers, suspended from a giant crane, observed the landings of Golden-crowned kinglets (Regulus strapa). They found that 78 of the 163 landings were in the "mid" height region of trees, 21-40 meters above ground. A) Compute an estimate of the population proportion of all landings of these kinglets that are in the 21-40 meter height range. B) Based on the data above, what is the margin of error of \hat{p} ?

What is the standard error of a statistic, and what information does it provide?

A recent study investigated the effects of a "Buckle Up Your Toddlers" campaign to get parents to use grocery cart seat belts. Investigators observed a representative sample of parents at grocery stores in a large city, and found 192 out of 594 parents buckling up their toddlers. A) Compute an estimate of the population proportion of all parents who buckle up their toddlers B) Based on the data above, what is the margin of error of $\hat { p }$ ?

What is the interpretation of the margin of error?

What does the confidence level tell you about a confidence interval?

In a couple sentences, define a "biased statistic" and "unbiased statistic." What is the difference between the two?

Briefly define the following terms in your own words: a) Confidence interval b) The confidence level associated with a confidence interval c) The margin of error associated with a confidence interval

A newspaper recently conducted a survey of a random sample of 120 registered voters in order to predict the winner of a local election. The Republican candidate was preferred by 62 of the respondents. Assuming that it is reasonable to regard the 120 voters in the sample as representative of the population of voters, construct and interpret a 90% confidence interval for the proportion of registered voters who prefer the Republican candidate.

Using an unbiased statistic guarantees that estimates will almost always be close to the true value.