Exam 7: Statistical Intervals Based on a Single Sample

A random sample of 10 observations was selected from a normal population distribution. The sample mean and sample standard deviations were 20 and 3.2, respectively. A 95% prediction interval for a single observation selected from the same population is

If the width of a confidence interval for $\mu$ is too wide when the population standard deviation $\sigma$ is known, which one of the following is the best action to reduce the interval width?

Determine the following: a. The 90^th percentile of the chi-squared distribution with $V$ = 12. b. The 10^th percentile of the chi-squared distribution with $V$ = 12. c. $P \left( 10.98 \leq \chi ^ { 2 } \leq 36.78 \right)$ where $\chi ^ { 2 }$ is a chi-squared rv with $V$ = 22. d. $P \left( \chi ^ { 2 } < 14.611 \text { or } \chi ^ { 2 } > 37.652 \right)$ where $\chi ^ { 2 }$ is a chi-squared rv with $V$ = 25

The z curve is often called the t curve with degrees of freedom equal to __________.

The formula used to construct approximately $100 ( 1 - \alpha ) \%$ confidence interval for a population proportion p when the sample size n is large enough $( n p \geq 10 \text { and } n ( 1 - p ) \geq 10 )$ is given by __________, where $\hat { p }$ is the sample proportion, and $\hat { q } = 1 - \hat { p }$

The upper limit of a 95% confidence interval for the variance $\sigma ^ { 2 }$ of a normal population using a sample of size n and variance value $s ^ { 2 }$ is given by:

If one wants to develop a 90% confidence interval for the mean $\mu$ of a normal population, when the standard deviation $\sigma$ is known, the confidence level is

A study of the ability of individuals to walk in a straight line reported that accompanying data on cadence (strides per seconds) for a sample of n - 20 randomly selected healthy men: .95 .81 .93 .95 .93 .86 1.05 .92 .85 .81 .92 .96 .92 1.00 .78 1.06 1.06 .96 .85 .92 A normal probability plot gives substantial support to the assumption that the population distribution of cadence is approximately normal. A descriptive summary of the data from MINITAB follows. Variable N Mean Median StDev SEMean Cadence 20 0.9255 0.9300 0.0809 0.0181 a. Calculate and interpret a 95% confidence interval for a population mean cadence. b. Calculate and interpret a 95% prediction interval for the cadence of a single individual randomly selected from this population. c. Calculate an interval that includes at least 99% of the cadences in the population distribution using a confidence level of 95%.

Suppose that an investigator believes that virtually all values in the population are between 38 and 70. The appropriate sample size for estimating the true population mean $\mu$ within 2 units with 95% confidence level is approximately

Determine the confidence level for each of the following large-sample one-sided confidence bounds: a. Upper bound: $\bar { x } + .84$ $\bar { x } + .84 s / \sqrt { n }$ b. Lower bound: $\bar { x } - 2.05 s / \sqrt { n }$ c. Upper bound: $\bar { x } + .67 s / \sqrt { n }$

By how much must the sample size n be increased if the width of the CI $\left( \bar { x } \pm z _ { \alpha n } \cdot \frac { \sigma } { \sqrt { n } } \right)$ is to be halved? If the sample size is increased by a factor of 25, what effect will this have on the width of the interval? Justify your assertions.

The chi-squared critical value, $\hat { \lambda } _ { \alpha , v } ^ { 2 }$ , denotes the number on the measurement axis such that __________ of the area under the chi-squared curve with __________ degrees of freedom lies to the __________ of $\hat { \lambda } _ { \alpha , v } ^ { 2 }$ .

If we think of the width of the confidence interval as specifying its precision or accuracy, then the confidence level (or reliability) of the interval is __________ related to its precision.

The results of a Wagner turbidity test performed on 15 samples of standard Ottawa testing sand were (in microamperes) 26.9 25.8 24.4 24.1 26.4 25.9 24.0 21.7 24.9 25.9 27.3 26.7 26.9 24.8 27.3 a. Is it plausible that this sample was selected from a normal population distribution? b. Calculate an upper confidence bound with confidence level 90% for the population standard deviation of turbidity.

Let $t _ { v }$ denote the density function curve for a t-distribution with $v$ degrees of freedom. As v __________, the spread of the corresponding $t_{J}$ curve decreases.

Let $X _ { 1 } , \ldots \ldots , X _ { n }$ be a random sample from a population having a mean $\mu$ and standard deviation $\sigma$ . Provided that n is large, the Central Limit Theorem (CLT) implies that $\bar { X }$ is __________ distributed.

Which of the following statements are false about the chi-squared distribution with $v$ degrees of freedom?

Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected, and the alcohol content of each bottle is determined. Let μ denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (8.0, 9.6). a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. b. Consider the following statement: There is a 95% chance that μ is between 8 and 9.6. Is this statement correct? Why or why not? c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 8.0 and 9.6. Is this statement correct? Why or why not? d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 955 interval ire repeated 100 times, 95 of the resulting intervals will include μ . Is this statement correct? Why or why not?

Which of the following expressions are true about a large-sample upper confidence bound for the population mean $\mu$ ?