Question 1

A recent survey indicates that the proportion of season ticket holders for the school hockey team who renew their seats is about 0.80. Using a 95% confidence interval and a margin of error of  0.025, what is the approximate size of the sample needed to estimate the true proportion who plan to renew their seats?

Accepted Answer

The formula for the sample size needed to estimate a population proportion with a specified margin of error and confidence level is:

$n = (z^2*p*q)/E^2$

where:

n = sample size
z = z-score corresponding to the desired confidence level (in this case, 1.96 for 95% confidence)
p = estimated proportion (in this case, 0.80)
q = 1 - p
E = margin of error (in this case, 0.025)

Substituting in the given values, we get:

$n = (1.96^2*0.80*0.20)/(0.025^2) \approx 983.78 \approx 984$

Therefore, we need a sample size of at least 984 to estimate the true proportion of season ticket holders who plan to renew their seats with a margin of error of 0.025 and 95% confidence. The nearest option is C (984), so that is the correct answer.

Question 2

If the population deviation  is known and we wish to estimate the population mean  with 90% confidence, what is the appropriate critical z-value to use?

Accepted Answer

The answer of If the population deviation  is known...

Question 3

If we wish to construct a 95% confidence interval estimate for the difference between two population proportions, what would the confidence level be?

Accepted Answer

A 95% confidence interval estimate corresponds to a confidence level of 0.95.

Question 4

A random sample of 400 students was surveyed to determine an estimate for the proportion of all students who had attended at least three football games. The estimate revealed that between 0.372 and 0.458 of all students attended. Given this information, which of the following is the approximate value of the confidence coefficient?

Accepted Answer

The answer of A random sample of 400 students was...

Question 5

If the population deviation  is known and we wish to estimate the population mean  with 95% confidence, which of the following would be the appropriate critical z-value to use?

Accepted Answer

To estimate the population mean with 95% confidence, we need to use the 95% confidence interval, which corresponds to a critical value of 1.96 for a normal distribution. This is because 95% of the area under a normal curve falls within 1.96 standard deviations of the mean. Therefore, choice C, with a critical value of 1.96, is the appropriate answer.

Question 6

To what does the term "confidence level" refer?

Accepted Answer

Confidence level refers to the percentage of interval estimates that can be expected to contain the actual value of the parameter being estimated when the same procedure of interval construction is used again and again.

Question 7

Suppose you wish to estimate the difference between two population means when the population variances are known. Which critical values of z can you use to develop the 90% confidence interval estimate?

Accepted Answer

For a 90% confidence interval, the critical z-value is 1.645, which corresponds to the middle 90% of the standard normal distribution.

Question 8

In order to construct a 95% confidence interval estimate for the difference between the means of two normally distributed populations, where the unknown population variances are assumed not to be equal, the following summary statistics were computed from two independent samples:  ,  ,  ,  ,  , and  . In this case, what is the upper confidence limit?

Accepted Answer

The answer of In order to construct a 95% confidence...

Question 9

When two independent random samples of sizes  and  have been selected from populations with means  and  and variances  and  , respectively, which of the following is a property of the sampling distribution of  ?

Accepted Answer

The answer of When two independent random samples of sizes...

Question 10

In developing a confidence interval estimate for the difference between two population means, which of the following will result from an increase in the size of the sample?

Accepted Answer

Increasing the sample size increases the precision of the estimate and reduces the standard error, resulting in a narrower confidence interval. A larger sample size does not have an effect on the critical z-value.

Question 11

Which of the following are possible options when estimating a population mean  , where the population standard deviation  is known?

Accepted Answer

The answer of Which of the following are possible options...

Question 12

A 95% confidence interval for the population proportion of professional tennis players who earn more than $2 million a year is found to be between 0.82 and 0.88. What was the approximate sample size used to obtain this information?

Accepted Answer

The sample size can be approximated using the formula: n = (Z^2 * p * q) / e^2, where Z is the z-score for the desired confidence level, p is the estimated proportion, q is 1 - p, and e is the margin of error. Plugging in the given values, we get: n = (1.96^2 * 0.85 * 0.15) / 0.03^2 = 544.64. The closest answer choice is A) 545.

Question 13

What is the z-value needed to construct a 92.5% confidence interval estimate for the difference between two population proportions?

Accepted Answer

The z-value for a 92.5% confidence interval is approximately 1.78, as it corresponds to the middle 92.5% of the standard normal distribution.

Question 14

A statistician wishes to reduce the margin of error associated with a confidence interval estimate for a population proportion p. What does she or he need to do?

Accepted Answer

The answer of A statistician wishes to reduce the margin...

Question 15

If a 90% confidence interval estimate for the difference between two population proportions is to be constructed, what would the confidence coefficient be?

Accepted Answer

The confidence coefficient is equal to 1 minus the level of significance or alpha, which is usually set at 0.10, 0.05, or 0.01. For a 90% confidence interval, the level of significance would be 0.10, so the confidence coefficient would be 1 - 0.10 = 0.90.

Question 16

What is the z-value needed to construct a 97.8% confidence interval estimate for the difference between two population proportions?

Accepted Answer

The z-value for a 97.8% confidence interval corresponds to the critical value that leaves 1.1% in each tail of the standard normal distribution, which is approximately 2.29.

Question 17

Suppose you wish to estimate the difference between two population means when the population variances are known. Which critical value of z can you use to develop the 99% confidence interval estimate?

Accepted Answer

A 99% confidence interval corresponds to a z-score of 2.575 (looked up in a standard normal distribution table or using a calculator).

Question 18

What is the approximate z-value you would use if you wish to construct an 85% upper confidence bound for the population proportion p?

Accepted Answer

For an 85% upper confidence bound, you need the z-value that corresponds to the 85th percentile of the standard normal distribution, which is approximately 1.04.

Question 19

What is the approximate z-value you would use if you wish to construct an 80% lower confidence bound for the population mean  ?

Accepted Answer

The answer of What is the approximate z-value you would...

Question 20

Suppose the population standard deviation equals 10. What is the sample size needed to estimate, with 95% confidence, a population mean within 1.5 units of its true value?

Accepted Answer

To find the sample size, use the formula n = (Z*σ/E)^2, where Z is the Z-score for 95% confidence (1.96), σ is the population standard deviation (10), and E is the margin of error (1.5). Plugging in these values gives n = (1.96*10/1.5)^2 ≈ 171.

Quiz 8: Large-Sample Estimation

If the population deviation is known and we wish to estimate the population mean with 90% confidence, what is the appropriate critical z-value to use?