Question 1

Consider a confidence interval for a population mean.As the standard deviation ________,sample sizes need to _____________ to achieve a specified margin of error.

Accepted Answer

As the standard deviation increases, the variability in the data increases, which means that we need larger sample sizes to allow for more accurate estimates of the population mean. This will result in a smaller margin of error, which means a narrower confidence interval, giving us more precision in our estimation.

Question 2

When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be narrower when $\hat { p } ( 1 - \hat { p } )$ 
is larger than when $\hat { p } ( 1 - \hat { p } )$ 
is smaller.

Accepted Answer

The width of a confidence interval for a population proportion p is inversely related to the square root of the sample size n and directly related to the standard deviation of the sampling distribution, which is maximized when $\hat { p } = 0.5$. Therefore, a larger $\hat { p } ( 1 - \hat { p } )$ (indicating a value of $\hat { p }$ closer to 0.5) actually results in a wider confidence interval, not a narrower one.

Question 3

Consider a normally distributed population.Which two of the following actions will result in the greatest increase in the width of a confidence interval for the mean?
(i)increasing the confidence level
(ii)decreasing the confidence level
(iii)increasing the sample size
(iv)decreasing the sample size
(v)decreasing the standard deviation

Accepted Answer

Increasing the confidence level (i) requires a wider interval to ensure the mean falls within that interval with higher certainty. Decreasing the sample size (iv) increases the standard error, which also widens the confidence interval.

Question 4

An injector molder produces plastic pens.The process is designed to produce pens with a mean weight of 0.250 grams.To investigate whether the injection molder is operating satisfactorily,40 pens were randomly sample with a mean of 0.2525g.The population standard deviation is believed to be approximately 0.0022g.Find a 99% confidence interval for the population mean.

Accepted Answer

The answer of An injector molder produces plastic pens.The process...

Question 5

Consider a normally distributed population with standard deviation 36.If we wish to estimate the population mean to within 10 units with 90% confidence,what is the required sample size?

Accepted Answer

The answer of Consider a normally distributed population with standard...

Question 6

Suppose we are interested in the population mean of a normally distributed population with unknown standard deviation.If we take a sample of 12 observations from this population,the confidence interval for the population mean is computed based on the t distribution.

Accepted Answer

This is because when the population standard deviation is unknown and we have a sample size of less than 30, we use the t-distribution to calculate the confidence interval for the population mean. Therefore, in this case, we would use the t-distribution to compute the confidence interval for the population mean based on the sample of 12 observations.

Question 7

If we wish to estimate a population proportion to within 5% with 95% confidence,the required sample size is 385.

Accepted Answer

This is true. The sample size required to estimate a population proportion with a certain level of precision and confidence is dependent on the margin of error, desired level of confidence, and population size. Using the formula n = (z^2 * p * (1-p)) / (e^2) where z is the z-score corresponding to the desired level of confidence, p is the estimated population proportion, and e is the desired margin of error, we can calculate that a sample size of 385 is required to estimate a population proportion to within 5% with 95% confidence.

Question 8

When the level of confidence and sample proportion $\hat { p }$ 
remain the same,a confidence interval for a population proportion p based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400.

Accepted Answer

The width of a confidence interval for a population proportion is inversely related to the square root of the sample size (n), meaning as the sample size increases, the confidence interval becomes narrower, assuming the level of confidence and sample proportion remain constant.

Question 9

We wish to compute a confidence interval for a population mean.Suppose we are determining the sample size n needed to achieve a desired margin of error for this interval.If we compute the value of n to be 79.2,we would choose to sample 79 observations.

Accepted Answer

Sample size cannot be a non-integer value, so we would choose to round up to the nearest whole number and sample 80 observations.

Question 10

A 95% confidence interval for a population proportion is computed to be [0.14,0.36].This means that each time we take a sample from this population,the sample proportion has a 95% chance of being between 0.14 and 0.36.

Accepted Answer

The correct interpretation of a 95% confidence interval is that if we were to take 100 different samples from the same population and compute the confidence interval for each sample, approximately 95 of those intervals would contain the true population proportion. We cannot say that any individual sample has a 95% chance of containing the population proportion, as it either does or does not contain the true value.

Question 11

When the population is normally distributed and the population standard deviation $\sigma$ 
is unknown,then for any sample size n,the confidence interval for $\mu$ 
is based on the z distribution.

Accepted Answer

When the population standard deviation $\sigma$ is unknown, the confidence interval for $\mu$ is based on the t-distribution, not the z-distribution, regardless of the sample size $n$.

Question 12

When the level of confidence and standard deviation remain the same,a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50.

Accepted Answer

This is because the standard error, which is the estimate of the standard deviation of the sampling distribution, decreases as sample size increases. A smaller standard error results in a narrower confidence interval. Therefore, a larger sample size leads to a more accurate estimate of the population mean and a narrower confidence interval, assuming that the level of confidence and standard deviation remain the same.

Question 13

All other things being constant,the margin of error for a confidence interval for a population mean increases as the sample size decreases.

Accepted Answer

This is because as the sample size decreases, there is less data to work with and the estimates may be less accurate. Therefore, the margin of error increases as a result.

Question 14

It is appropriate to compute a confidence interval for a population proportion p if $n \hat { p }$ 
and $n ( 1 - \hat { p } )$ 
are both at least 5.

Accepted Answer

This rule ensures that the sample size is large enough for the normal approximation to the binomial distribution to be appropriate, which is a key assumption for computing confidence intervals for population proportions.

Question 15

Consider a normally distribution population with known variance.The width of a 99% confidence interval for the population mean will be _______ the width of a 95% confidence interval for the population mean:

Accepted Answer

The width of a 99% confidence interval is greater than the width of a 95% confidence interval because a higher confidence level requires a wider interval to ensure that it encompasses the population mean with greater certainty.

Question 16

Assuming the same confidence level $1 - \alpha$ 
,as the sample size increases,the value of $t _ { \alpha / 2 }$ 
approaches the value of $z _ { \alpha / 2 }$ 
.

Accepted Answer

As the sample size increases, the t-distribution approaches the normal distribution, making $t _ { \alpha / 2 }$ approach $z _ { \alpha / 2 }$ due to the Central Limit Theorem.

Question 17

For a fixed sample size and standard deviation,the width of a confidence interval for a population mean $\mu$ 
will increase as the level of confidence decreases.

Accepted Answer

The width of a confidence interval for a population mean $\mu$ will decrease as the level of confidence decreases, because a lower confidence level means you require less certainty about where the true parameter lies, resulting in a narrower interval.

Question 18

When constructing a confidence interval for a sample proportion,the t distribution is appropriate if the sample size is small.

Accepted Answer

The t distribution is not appropriate for constructing confidence intervals for sample proportions. Instead, the normal distribution or the z distribution is used for larger sample sizes (n > 30). For smaller sample sizes (n < 30), a modified version of the normal distribution called the t distribution can be used. However, the t distribution is not specifically used for sample proportions but instead for other types of data such as sample means or differences in means.

Question 19

When constructing a confidence interval for a population mean,if the population is normally distributed with unknown standard deviation and the sample size is 10,then the distribution of $\bar { x }$
Is based on _____ distribution.

Accepted Answer

When the population standard deviation is unknown and the sample size is small (n < 30), the distribution of the sample mean $\bar{x}$ is best approximated using the t-distribution, not the z-distribution. The t-distribution accounts for the increased uncertainty due to estimating the population standard deviation with the sample standard deviation.

Question 20

When computed based on the same sample,the 99% confidence interval for a population mean $\mu$ 
will be narrower than the 95% confidence interval for $\mu$ 
.

Accepted Answer

A 99% confidence interval is wider than a 95% confidence interval because it requires capturing more of the data to be more confident about the estimate, thus extending the range.

Quiz 9: Statistical Inferences Based on Two Samples

Consider a confidence interval for a population mean.As the standard deviation ,sample sizes need to _____ to achieve a specified margin of error.

When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be narrower when $\hat { p } ( 1 - \hat { p } )$ is larger than when $\hat { p } ( 1 - \hat { p } )$ is smaller.

Consider a normally distributed population with standard deviation 36.If we wish to estimate the population mean to within 10 units with 90% confidence,what is the required sample size?

Suppose we are interested in the population mean of a normally distributed population with unknown standard deviation.If we take a sample of 12 observations from this population,the confidence interval for the population mean is computed based on the t distribution.

If we wish to estimate a population proportion to within 5% with 95% confidence,the required sample size is 385.

When the level of confidence and sample proportion $\hat { p }$ remain the same,a confidence interval for a population proportion p based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400.

We wish to compute a confidence interval for a population mean.Suppose we are determining the sample size n needed to achieve a desired margin of error for this interval.If we compute the value of n to be 79.2,we would choose to sample 79 observations.

A 95% confidence interval for a population proportion is computed to be [0.14,0.36].This means that each time we take a sample from this population,the sample proportion has a 95% chance of being between 0.14 and 0.36.

When the population is normally distributed and the population standard deviation $\sigma$ is unknown,then for any sample size n,the confidence interval for $\mu$ is based on the z distribution.

When the level of confidence and standard deviation remain the same,a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50.

All other things being constant,the margin of error for a confidence interval for a population mean increases as the sample size decreases.

It is appropriate to compute a confidence interval for a population proportion p if $n \hat { p }$ and $n ( 1 - \hat { p } )$ are both at least 5.

Consider a normally distribution population with known variance.The width of a 99% confidence interval for the population mean will be _______ the width of a 95% confidence interval for the population mean:

Assuming the same confidence level $1 - \alpha$ ,as the sample size increases,the value of $t _ { \alpha / 2 }$ approaches the value of $z _ { \alpha / 2 }$ .

For a fixed sample size and standard deviation,the width of a confidence interval for a population mean $\mu$ will increase as the level of confidence decreases.

When constructing a confidence interval for a sample proportion,the t distribution is appropriate if the sample size is small.

When constructing a confidence interval for a population mean,if the population is normally distributed with unknown standard deviation and the sample size is 10,then the distribution of $\bar { x }$ Is based on _____ distribution.

When computed based on the same sample,the 99% confidence interval for a population mean $\mu$ will be narrower than the 95% confidence interval for $\mu$ .

Quiz 9: Statistical Inferences Based on Two Samples

Consider a confidence interval for a population mean.As the standard deviation ________,sample sizes need to _____________ to achieve a specified margin of error.

When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be narrower when p^(1−p^)\hat { p } ( 1 - \hat { p } )p^​(1−p^​) is larger than when p^(1−p^)\hat { p } ( 1 - \hat { p } )p^​(1−p^​) is smaller.

Consider a normally distributed population with standard deviation 36.If we wish to estimate the population mean to within 10 units with 90% confidence,what is the required sample size?

Suppose we are interested in the population mean of a normally distributed population with unknown standard deviation.If we take a sample of 12 observations from this population,the confidence interval for the population mean is computed based on the t distribution.

If we wish to estimate a population proportion to within 5% with 95% confidence,the required sample size is 385.

When the level of confidence and sample proportion p^\hat { p }p^​ remain the same,a confidence interval for a population proportion p based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400.

We wish to compute a confidence interval for a population mean.Suppose we are determining the sample size n needed to achieve a desired margin of error for this interval.If we compute the value of n to be 79.2,we would choose to sample 79 observations.

A 95% confidence interval for a population proportion is computed to be [0.14,0.36].This means that each time we take a sample from this population,the sample proportion has a 95% chance of being between 0.14 and 0.36.

When the population is normally distributed and the population standard deviation σ\sigmaσ is unknown,then for any sample size n,the confidence interval for μ\muμ is based on the z distribution.

When the level of confidence and standard deviation remain the same,a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50.

All other things being constant,the margin of error for a confidence interval for a population mean increases as the sample size decreases.

It is appropriate to compute a confidence interval for a population proportion p if np^n \hat { p }np^​ and n(1−p^)n ( 1 - \hat { p } )n(1−p^​) are both at least 5.

Consider a normally distribution population with known variance.The width of a 99% confidence interval for the population mean will be _______ the width of a 95% confidence interval for the population mean:

Assuming the same confidence level 1−α1 - \alpha1−α ,as the sample size increases,the value of tα/2t _ { \alpha / 2 }tα/2​ approaches the value of zα/2z _ { \alpha / 2 }zα/2​ .

For a fixed sample size and standard deviation,the width of a confidence interval for a population mean μ\muμ will increase as the level of confidence decreases.

When constructing a confidence interval for a sample proportion,the t distribution is appropriate if the sample size is small.

When constructing a confidence interval for a population mean,if the population is normally distributed with unknown standard deviation and the sample size is 10,then the distribution of xˉ\bar { x }xˉ Is based on _____ distribution.

When computed based on the same sample,the 99% confidence interval for a population mean μ\muμ will be narrower than the 95% confidence interval for μ\muμ .

Consider a confidence interval for a population mean.As the standard deviation ,sample sizes need to _____ to achieve a specified margin of error.

When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be narrower when $\hat { p } ( 1 - \hat { p } )$ is larger than when $\hat { p } ( 1 - \hat { p } )$ is smaller.

When the level of confidence and sample proportion $\hat { p }$ remain the same,a confidence interval for a population proportion p based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400.

When the population is normally distributed and the population standard deviation $\sigma$ is unknown,then for any sample size n,the confidence interval for $\mu$ is based on the z distribution.

It is appropriate to compute a confidence interval for a population proportion p if $n \hat { p }$ and $n ( 1 - \hat { p } )$ are both at least 5.

Assuming the same confidence level $1 - \alpha$ ,as the sample size increases,the value of $t _ { \alpha / 2 }$ approaches the value of $z _ { \alpha / 2 }$ .

For a fixed sample size and standard deviation,the width of a confidence interval for a population mean $\mu$ will increase as the level of confidence decreases.

When constructing a confidence interval for a population mean,if the population is normally distributed with unknown standard deviation and the sample size is 10,then the distribution of $\bar { x }$ Is based on _____ distribution.

When computed based on the same sample,the 99% confidence interval for a population mean $\mu$ will be narrower than the 95% confidence interval for $\mu$ .