Question 1

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 2

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

Accepted Answer

The margin of error in a confidence interval is affected by several factors, including the sample size and the level of confidence. The margin of error is typically calculated as a multiple of the standard error, which is based on the sample size and the variability of the data. A 95 percent confidence interval estimate will have a margin of error that is approximately ± 1.96 times the standard error, but this value is not directly related to the size of the population mean.

Question 3

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 cannot be determined solely based on the sample mean and standard deviation. Information regarding the sample size or confidence level is also needed.

Question 4

All other factors held constant,the higher the confidence level,the closer the point estimate for the population mean will be to the true population mean.

Accepted Answer

The answer of All other factors held constant,the higher the...

Question 5

A 95 percent confidence interval for a mean will contain 95 percent of the population within the interval.

Accepted Answer

A 95 percent confidence interval refers to the likelihood that the sample mean falls within the given interval, not a guarantee that 95 percent of the population falls within that same interval.

Question 6

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

The upper limit for the 90 percent confidence interval estimate for the population mean can be calculated as:
Upper limit = sample mean + t-value*(standard error)
Assuming a 90 percent confidence interval, the t-value with 8 degrees of freedom is approximately 1.86. The standard error can be calculated as standard deviation/(sqrt(n)), which is equal to 3.5/(sqrt(9)) = 1.17. Substituting these values, we get:
Upper limit = 15.3 + 1.86*(1.17) ≈ 17.47
Therefore, the statement is true.

Question 7

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 8

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

Accepted Answer

While some factors such as the level of confidence and margin of error are within the control of the decision maker, other factors such as the variability of the population and the effect size may not be as easily controlled. Therefore, the required sample size may not always be entirely within the control of the decision maker.

Question 9

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 upper limit for a 95 percent confidence interval estimate is approximately 12.106 ounces.

Accepted Answer

The upper limit for a 95% confidence interval estimate can be calculated as:

Upper limit = sample mean + (critical value x standard error)

The critical value for a 95% confidence interval with 11 degrees of freedom (12 bottles in the sample minus 1) is 2.201.

The standard error can be calculated as the standard deviation of the sample divided by the square root of the sample size:

Standard error = s / sqrt(n) = 0.12 / sqrt(12) = 0.0346.

Therefore, the upper limit can be estimated as:

Upper limit = 12.03 + (2.201 x 0.0346) = 12.1058 ~ 12.106

So the statement is true, and the upper limit for a 95% confidence interval estimate is approximately 12.106 ounces.

Question 10

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

Accepted Answer

The t-distribution can be used with non-normal populations, but there may be some limitations or considerations to take into account.

Question 11

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 leads to a smaller standard error, which results in a smaller margin of error around the estimated population mean. Therefore, a large sample size is generally preferred for estimating a population mean.

Question 12

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

When the population standard deviation is unknown, the sample standard deviation is used as an estimate, and the t-distribution is used to calculate the critical value for the confidence interval. The normal distribution is used when the population standard deviation is known.

Question 13

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 is one of the key factors in determining the sample size needed for a confidence interval estimate. The higher the desired level of confidence, the larger the sample size needed to achieve that level of precision.

Question 14

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 15

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

Accepted Answer

This statement is true. The margin of error is the amount added and subtracted from a point estimate in order to create a confidence interval. The width of the confidence interval is therefore twice the margin of error, which means that the margin of error is one-half the width of the confidence interval.

Question 16

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 the population mean based on a sample from a population for which we do not know the population standard deviation, we use a t-distribution to calculate a confidence interval for the population mean. Therefore, the critical value will be a t-value from the t-distribution.

Question 17

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

Accepted Answer

This statement is true. The t-distribution is used when the sample size is small and/or the population distribution is not known. It provides critical values for constructing confidence intervals and conducting hypothesis tests.

Question 18

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

This is because smaller sample sizes have less data points to work with, leading to a larger standard error and a wider distribution of potential values. As a result, the confidence interval must be wider in order to accommodate this greater uncertainty.

Question 19

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 20

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 is dependent on the level of confidence and the sample size, but not directly on the point estimate.

Quiz 8: Estimating Single Population Parameters