Exam 9: Estimation Using a Single Sample

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

$\hat{p}$ is an estimator.

Under what conditions is it reasonable to use the z confidence interval to estimate apopulation proportion?

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

All else being equal, the higher the confidence level, the wider theconfidence interval.

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

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

In a very large school district, student records have been kept in three distinct databases, one each for elementary, middle, and high school. The district now has a newdata base and wishes to correct any original data entry errors. To allocate enoughsecretarial time to this task, an estimate of the proportion of records that containerrors must be made. Initial estimates are that there are fixable errors in about 15% ofthe records.a) Using 15% as an initial estimate of the percentage of records with errors, computethe sample size required to estimate the true proportion of records with errorswithin .025 with 95% confidence.b) Suppose the school district did not have an initial estimate of p. In a fewsentences, explain how your procedure for choosing a samples size would differfrom your solution in part (a). (Do NOT recalculate a new estimate of thenecessary sample size!)

The width of the confidence interval for p decreases as the sample sizeincreases.

Eyestrain is thought to be associated with different types of office work. The AcmeTemp-Help Company provides short-term employees to substitute for vacationingdata entry workers. Previous research has shown that approximately 21% of full timedata 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 acceptthe value of 0.21 for full-time data entry workers as a reasonable initial estimateof p, what sample size should they use for their study? b) Suppose Acme believed that the working conditions of their temporary data entrypersonnel are different enough that they should not depend on the 0.21 as aninitial estimate. In a few sentences, explain how your procedure for choosing asamples size would differ from your solution in part (a). (Do NOT recalculate anew estimate of the necessary sample size!)

When conducting research into the efficacy of drugs to combat psychologicaldisorders it is very common to use double blinding as a design strategy. In a doubleblind study neither the patient nor the doctor knows whether the patient is receivingthe treatment or the placebo drug. Does double blinding actually work? That is, isthe 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 patientswith panic attack disorder and agoraphobia were treated. After completing thestudy the physicians involved were asked to guess the treatments. The physicianscorrectly guessed the treatments for 30 patients. Determine whether or not thesample size is large enough to use the standard normal distribution to find themargin of error. Be sure to show any supporting calculations.In a different study of double-blinding, the use of lithium as a treatment to counteractaggressive tendencies in adolescents was under study. In that study the physicianscorrectly guessed the treatment in 37 out of 57 patients. b) Assuming these adolescents and doctors to be representative, estimate thepopulation proportion of patients for whom the doctor could correctly guess thetreatment. c) If the physicians were simply guessing about the treatments administered to thepatients, p = 0.50 would be the proportion of correct guesses. If the physicianswere guessing, what would be the standard error of $\hat{p}$ ? d) Based on the given data for the lithium treatment, calculate a 95% confidenceinterval for p, the proportion of correct guesses by physicians. e) Suppose the researchers wished to get a more precise estimate of the proportion ofcorrect guesses of treatment when using lithium in this context. What samplesize would be needed to obtain a margin of error of 0.05 with 95% confidence?

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

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

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

What is the interpretation of the margin of error?

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

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

The standard error of

A z of 1.96 is used when calculating 95% large-sample confidenceintervals for p.

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