Question 1

A higher confidence level means that the point estimate for the population mean will be closer to the true population mean.

Accepted Answer

A higher confidence level increases the width of the confidence interval, indicating a wider range of values that might contain the true population mean, but it does not guarantee that the point estimate is closer to the true population mean.

Question 2

The standard deviation for the checking account balances is assumed known to be $357.50. Recently, a bank manager was interested in estimating the mean balance. To do this, she selected a random sample of 81 accounts and found a mean balance of $1,347.20. At a 95 percent confidence level, the lower limit for the confidence interval is $646.50.

Accepted Answer

The calculation for the lower limit of a confidence interval for the mean, assuming the standard deviation is known, is given by $\bar{x} - Z \frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $Z$ is the Z-value corresponding to the confidence level (for 95%, $Z = 1.96$), $\sigma$ is the population standard deviation, and $n$ is the sample size. Given $\bar{x} = 1347.20$, $\sigma = 357.50$, and $n = 81$, the calculation for the lower limit does not result in $646.50.

Question 3

A pilot sample is one that is used when a decision maker wishes to get an advance idea of what the mean of the population might be.

Accepted Answer

The answer of A pilot sample is one that is...

Question 4

If the population is not normally distributed, the t-distribution cannot be used.

Accepted Answer

The t-distribution can still be used for small sample sizes from populations that are not normally distributed, especially as the sample size increases, due to the Central Limit Theorem which states that the distribution of the sample means approaches a normal distribution as the sample size becomes larger.

Question 5

The makers of weight loss product are interested in estimating the mean weight loss for users of their product. To do this, they have selected a random sample of n = 9 people and have provided them with a supply of the product. After six months, the nine people had an average weight loss of 15.3 pounds with a standard deviation equal to 3.5 pounds. The upper limit for the 90 percent confidence interval estimate for the population mean is approximately 17.47 pounds.

Accepted Answer

Given the sample size (n = 9), sample mean ($\bar{x}$ = 15.3), sample standard deviation (s = 3.5), and confidence level (90%), the upper limit of the confidence interval can be calculated using the t-distribution (due to the small sample size) and the formula for the confidence interval. The calculation involves finding the appropriate t-value for the degrees of freedom (n-1 = 8) and the specified confidence level, then computing the margin of error by multiplying the t-value by the standard error of the mean (s/√n). Adding this margin of error to the sample mean gives the upper limit of the confidence interval, which can indeed approximate to 17.47 pounds, making the statement true.

Question 6

All of the factors that are needed to determine the required sample size are within the control of the decision maker.

Accepted Answer

Some factors influencing the required sample size, such as the population variability and the desired confidence level, are not directly controllable by the decision maker. Population variability is an inherent characteristic of the population being studied, and while the confidence level can be chosen, it is constrained by practical considerations of accuracy and cost.

Question 7

If we are interested in estimating the population mean based on a sample from a population for which we know neither the mean nor the standard deviation, the critical value will be a t value from the t- distribution.

Accepted Answer

When estimating a population mean from a sample without knowing the population standard deviation, the t-distribution is used because it accounts for the added uncertainty of estimating the population standard deviation from the sample.

Question 8

In developing a confidence interval estimate, the margin of error is directly dependent on the value of the point estimate.

Accepted Answer

The margin of error in a confidence interval estimate primarily depends on the standard error of the statistic and the critical value associated with the confidence level, not directly on the value of the point estimate itself.

Question 9

A 95 percent confidence interval estimate will have a margin of error that is approximately ±47.5 percent of the size of the population mean.

Accepted Answer

A 95 percent confidence interval estimate's margin of error is determined by the standard error of the mean and the critical value from the z-distribution or t-distribution, not a percentage of the population mean.

Question 10

One factor that plays an important part in determining what the needed sample size is when developing a confidence interval estimate is the level of confidence that you wish to use.

Accepted Answer

The level of confidence determines the width of the confidence interval; higher confidence levels require larger sample sizes to achieve the same width of the confidence interval.

Question 11

The bottlers of a new fruit juice daily select a random sample of 12 bottles of the drink to estimate the mean quantity of juice in the bottles filled that day. On one such day, the following results were observed:  Based on this information, the upper limit for a 95 percent confidence interval estimate is approximately 12.106 ounces.

Accepted Answer

The answer of The bottlers of a new fruit juice...

Question 12

One way to reduce the margin of error in a confidence interval estimate is to lower the level of confidence.

Accepted Answer

Lowering the level of confidence decreases the width of the confidence interval, thereby reducing the margin of error. This is because a lower confidence level means you are accepting a higher probability that the true parameter will not be captured by the interval, allowing the interval itself to be narrower.

Question 13

The impact on the margin of error for a confidence interval for an increase in confidence level and a decrease in sample size is unknown since these changes are contradictory.

Accepted Answer

Increasing the confidence level increases the margin of error, while decreasing the sample size also increases the margin of error. Both actions lead to a larger margin of error, not an unknown effect.

Question 14

The margin of error is one-half the width of the confidence interval.

Accepted Answer

The margin of error is defined as one-half the width of the confidence interval because the confidence interval is centered around the sample mean and extends equally in both directions (above and below the mean) by the margin of error.

Question 15

The bottlers of a new fruit juice daily select a random sample of 12 bottles of the drink to estimate the mean quantity of juice in the bottles filled that day. On one such day, the following results were observed: = 12.03; s = 0.12. Based on this information, the margin of error associated with a 90 percent confidence interval estimate for the population mean is 1.7959 ounces.

Accepted Answer

The margin of error for a confidence interval is calculated using the formula $ME = t_{\alpha/2} \times \frac{s}{\sqrt{n}}$, where $t_{\alpha/2}$ is the t-score corresponding to the desired confidence level, $s$ is the sample standard deviation, and $n$ is the sample size. The given values do not provide enough information to directly confirm the margin of error as 1.7959 ounces without knowing the t-score for the 90% confidence level and performing the calculation.

Question 16

A 95 percent confidence interval for a mean can be interpreted that we are 95 percent sure that our answer is correct.

Accepted Answer

A 95 percent confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect 95 of those intervals to contain the true population mean. It does not mean we are 95 percent sure our specific interval contains the true mean.

Question 17

When calculating a confidence interval, the reason for using the t-distribution rather than the normal distribution for the critical value is that the population standard deviation is unknown.

Accepted Answer

The t-distribution is used instead of the normal distribution when the population standard deviation is unknown because it accounts for the additional uncertainty introduced by estimating the standard deviation from the sample.

Question 18

The t-distribution is used to obtain the critical value in developing a confidence interval when the population distribution is not known or the sample size is small.

Accepted Answer

The t-distribution is used instead of the normal distribution to calculate the critical value for developing a confidence interval when the population standard deviation is unknown or the sample size is small (typically n < 30).

Question 19

Confidence intervals constructed with small samples tend to have greater margins of error than those constructed from larger samples, all else being constant.

Accepted Answer

Larger samples provide more information and tend to produce more precise estimates, leading to narrower confidence intervals with smaller margins of error compared to smaller samples.

Question 20

In estimating a population mean, a large sample size is generally preferable to a small sample size because the margin of error is generally smaller.

Accepted Answer

A larger sample size reduces the margin of error because it provides a more precise estimate of the population mean, leading to narrower confidence intervals.

Quiz 8: Estimating Single Population Parameters