Question 1

The width of the one-sample confidence interval for μ decreases as the sample size grows larger.

Accepted Answer

As the sample size grows larger, the standard error of the mean decreases, leading to a smaller margin of error and a narrower confidence interval.

Question 2

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 personnel. Previous research has shown that approximately 21% of full time data entry personnel have eyestrain, and Acme would like to estimate the proportion, π, of their 6,000 part-time data entry employees that have eyestrain. 
a) They would like to estimate π to within 0.05 with 95% confidence. If they accept the starting value of 0.21 as a reasonable initial estimate of π, what sample size should they use for their study? 
b) Suppose Acme had felt that the working conditions of their temporary data entry personnel were 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!)

Accepted Answer

a)   = 254.93 Thus, a sample of at least 255 will be needed. 
b) If no prior estimate of the population proportion is available, using p = 0.50 in the formula for n above would provide a sufficient sample size.

Question 3

Conventional home smoke alarms have too often failed to awaken sleeping children--they are remarkably resistant to being awakened by sound, even in the daytime! Researchers are studying the possibility that a personalized "voice" smoke alarm would be more effective than mere sound. The idea is that a recording of the mother's voice calling the child's name may be an improvement over existing systems. 
a) The researchers would like to estimate π to within 0.05 with 95% confidence. Past experience has shown that conventional alarms are effective about 0.58 of the time. If this is accepted as a reasonable initial estimate of π, what sample size should they use for their study? 
b) Suppose the researchers felt that the original study, done with children of parents working in a single hospital, was different enough so that their results (p = 0.58) should not be relied upon. 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!)

Accepted Answer

a)   = 374.3255 Thus, a sample of at least 375 children will be needed. 
b) If no price estimate of the population proportion is available, using p = 0.50 in the formula for n above would provide a sufficient sample size.

Question 4

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

Accepted Answer

This statement is true as a higher confidence level indicates greater certainty in the accuracy of the estimate, which requires a wider margin of error to account for any possible variation in the sample data.

Question 5

Under what conditions is it reasonable to use the t confidence interval to estimate a population mean?

Accepted Answer

The answer of Under what conditions is it reasonable to...

Question 6

An understanding of bird's choices of habitat is a crucial element in understanding avian biology and ecology. One aspect of habitat choice is the pattern of their 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, 21-40 meters above ground. 
a) Compute a point estimate of the population proportion of all landings of these kinglets that are in the 21 - 40 meter height range. 
b) Construct and interpret a 95% confidence interval for π, the population proportion of all landings of these kinglets that are in the 21 - 40 meter height range.

Accepted Answer

The answer of An understanding of bird's choices of habitat...

Question 7

Since s² gives an unbiased estimate of σ², s gives an unbiased estimate of σ.

Accepted Answer

The notation seems to be a mix-up or incorrectly formatted, but generally, $s$ (the sample standard deviation) does not give an unbiased estimate of $\sigma$ (the population standard deviation). To estimate $\sigma$ unbiasedly, a correction factor is typically applied to $s$.

Question 8

In a study of termite populations in Swaziland, the following data were gathered on the percentage of termites that were classified as "major workers" at eight randomly selected sites: 37.6 20.0 28.8 29.9 51.9 13.6 34.5 29.0 a) Use the given data to produce a point estimate of μ, the true mean percentage of termites that are "major workers." b) Use the given data to produce an unbiased point estimate of σ², the population variance of the percentage of termites that are "major workers."

Accepted Answer

The answer of In a study of termite populations in...

Question 9

The standard error of p is  .

Accepted Answer

The given code appears to be a unique identifier and is not related to the standard error of p. The standard error of p is a statistical measurement that represents the variability of sample proportions in relation to the population mean. It is typically calculated using the formula: standard error of p = sqrt[(p(1-p))/n], where p is the sample proportion and n is the sample size.

Question 10

Under what conditions is it reasonable to use the z confidence interval to estimate a population mean?

Accepted Answer

The answer of Under what conditions is it reasonable to...

Question 11

Using an unbiased statistic guarantees that there will be no systematic tendency to under- or overestimate the value of the population characteristic and that estimates will almost always be relatively close to the true value.

Accepted Answer

An unbiased statistic means that the expected value of the statistic equals the true population parameter, but it does not guarantee that individual estimates will be close to the true value; variability or sampling error can still result in individual estimates being far from the true value.

Question 12

The confidence interval formula for estimating μ, that is used when n is large, is based on the Central Limit Theorem.

Accepted Answer

The Central Limit Theorem states that when sample sizes are sufficiently large, the sampling distribution of the mean will be approximately normally distributed. This allows us to use the formula for a confidence interval, which is based on the assumption of normality.

Question 13

If the distribution of a population is symmetric, both the mean and median give unbiased estimates of the population mean. What criteria would you consider in choosing which statistic to use if you wanted to estimate the population mean?

Accepted Answer

The answer of If the distribution of a population is...

Question 14

The large sample confidence interval formula for estimating π can always safely be used if n ≥ 30.

Accepted Answer

The large sample confidence interval formula for estimating a population proportion π assumes not only that the sample size n is large (often n ≥ 30 is used as a rule of thumb) but also that the sample proportion is not too close to 0 or 1, specifically that both np and n(1-p) are greater than or equal to 10.

Question 15

A t curve is bell-shaped like the z curve but is less spread out.

Accepted Answer

A t curve is bell-shaped like the z curve but is more spread out, especially with smaller sample sizes, reflecting greater variability.

Question 16

The standard error of  is s.

Accepted Answer

A standard error is a measure of the variability of a statistic, usually the sample mean. It cannot be calculated for a single data point.

Question 17

A recent study investigated the effects of a "Buckle Up Your Toddlers" campaign to get parents to use the grocery cart seat belts. Investigators observed a simple random sample of parents in a large city, and found 192 out of 594 parents buckling up their toddlers. 
a) Compute a point estimate of the true proportion of all parents who buckle up their toddlers. 
b) Construct and interpret a 95% confidence interval for π, the true proportion of all parents who buckle up their toddlers.

Accepted Answer

The answer of A recent study investigated the effects of...

Question 18

What is the difference between an unbiased statistic and a biased statistic?

Accepted Answer

The answer of What is the difference between an unbiased...

Question 19

The t confidence interval formula for estimating μ should only be used when the population being sampled is at least approximately normally distributed.

Accepted Answer

The answer of The t confidence interval formula for estimating...

Question 20

Bicycle crashes are a common cause of severe injuries in childhood. A researcher theorized that this might be because of inadequate "gap" time (the time between a bicycling child clearing an intersection and the subsequent entry of an oncoming car into that intersection). A researcher experimentally investigated the gap judgments of 10-year-olds by placing them in a computer-generated virtual environment of cars approaching an intersection at 25 mph. The following gap times were observed for ten randomly selected children: 6.72, 4.18, 2.32, 4.05, 8.39, 5.44, 6.24, 2.05, 3.29, 5.38 a) Use the given data to produce a point estimate of μ, the population mean gap time of 10-year-olds from this population. b) Use the given data to produce an unbiased point estimate of σ², the population variance of the gap time of 10-year-olds from this population.

Accepted Answer

The answer of Bicycle crashes are a common cause of...

Quiz 9: Estimation Using a Single Sample