Question 1

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 size is small (less than 30), and the population is assumed to be normally distributed, the t-distribution is used to estimate the critical value for the confidence interval calculation.

Question 2

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

Using the formula for a confidence interval with a sample size of 12 and a 95% confidence level, we get:
Upper limit = x̄ + t(0.025,11)*s/sqrt(n)
Upper limit = 12.03 + 2.201*0.12/sqrt(12)
Upper limit = 12.106
Therefore, the given statement is true.

Question 3

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 that affect sample size, such as variance and effect size, may be outside of the decision maker's control. However, steps can be taken to minimize their impact and make informed decisions about sample size.

Question 4

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 for a mean means that if we repeated the sampling process and created an interval estimate for each sample, then 95 percent of those intervals would contain the population mean. It does not necessarily mean that 95 percent of the population falls within that interval. The population mean is a single value and is not subject to being "contained" within an interval.

Question 5

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

This statement is true. The t-distribution is commonly used when the sample size is small or the population standard deviation is unknown, making it difficult to use the normal distribution. The critical value from the t-distribution is used to calculate the confidence interval.

Question 6

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 7

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

Accepted Answer

The t-distribution is designed to be used for small sample sizes where the population distribution may not be known or may not be normal. So, it can be used even if the population is not normally distributed.

Question 8

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 as the sample size increases, the standard error of the mean decreases, resulting in a smaller margin of error for the confidence interval.

Question 9

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 10

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

We can use the formula for a confidence interval to find the margin of error, which is given by:

Margin of Error = z* (standard deviation / square root of sample size)

At a 95% confidence level, the z-value is 1.96. Plugging in the given values, we get:

Margin of Error = 1.96 * (357.50 / sqrt(81)) = 97.88

To find the lower limit of the confidence interval, we subtract the margin of error from the sample mean:

Lower Limit = Sample Mean - Margin of Error = 1347.20 - 97.88 = 1249.32

Since the lower limit in the question is given as $646.50, which is much lower than our calculated lower limit of $1249.32, the answer must be False.

Question 11

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

Accepted Answer

This is true. The margin of error represents how much the sample mean may deviate from the true population mean. The confidence interval provides a range of values around the sample mean that is likely to contain the true population mean. The width of this interval is determined by the level of confidence and the standard error of the sample mean. Therefore, the margin of error is equal to one-half the width of the confidence interval.

Question 12

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

Question 13

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 the population mean and standard deviation are unknown and the sample size is small (n<30), we use the t-distribution and obtain the critical value from a t-table.

Question 14

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

With a larger sample size, the sample mean becomes a more reliable estimate of the true population mean, resulting in a smaller margin of error. This is due to the central limit theorem, which states that as sample size increases, the distribution of sample means approaches a normal distribution.

Question 15

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

A higher confidence level results in a wider confidence interval, not necessarily a closer point estimate to the true population mean.

Question 16

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

Using the formula for calculating a confidence interval for the population mean, and given a sample size of 9, a sample mean of 15.3, a sample standard deviation of 3.5, and a confidence level of 90%, the upper limit for the confidence interval can be calculated as:

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

The critical value for a 90% confidence level with 8 degrees of freedom (9-1) is 1.860, which can be looked up in a t-distribution table. The standard error can be calculated as the standard deviation of the sample divided by the square root of the sample size:

SE = 3.5 / sqrt(9) = 1.167

Thus, the upper limit for the confidence interval is:

Upper limit = 15.3 + (1.860 x 1.167) = 17.47

Therefore, the statement is true.

Question 17

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 factors that determines the needed sample size for developing a confidence interval estimate. Other factors may include the variability of the population and the desired margin of error.

Question 18

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 19

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 for a confidence interval is based on the sample size, the standard deviation or variability of the population, and the level of confidence desired (such as 95%). The margin of error does not depend on the size of the population mean. Therefore, the statement in the question is false.

Question 20

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.

Quiz 8: Estimating Single Population Parameters