Exam 7: Statistical Intervals Based on a Single Sample

Which of the following statements are true about the percentiles of a chi-squared distribution with 20 degrees of freedom?

Let $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { n }$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma ^ { 2 }$ . Then the random variable $( n - 1 ) S ^ { 2 } / \sigma ^ { 2 }$ has a __________ probability distribution with __________ degrees of freedom.

It was reported that, in a sample of 507 adult Americans, only 142 correctly described the Bill of Rights as the first ten amendments to the U.S. Constitution. Calculate a (two-sided) confidence interval using a 99% confidence level for the proportion of all U. S. adults that could give a correct description of the Bill of Rights.

A random sample of n = 8 E-glass fiber test specimens of a certain type yielded a sample mean interfacial shear yield stress of 30.5 and a sample standard deviation of 3.0. Assuming that interfacial shear yield stress is normally distributed, compute a 95% CI for true average stress.

A random sample of 64 observations produced a mean value of 82 and standard deviation of 5.5. The 90% confidence interval for the population mean $\mu$ is between

If you want to develop a 99% confidence interval for the mean $\mu$ of a normal population, when the standard deviation $\sigma$ is known, the confidence level is __________.

Which of the following statements are true?

Which of the following statements are not true in developing a confidence interval for the population mean $\mu$

Which of the following statements are not true?

The superintendent of a large school district, having once had a course in probability and statistics, believes that the number of teachers absent on any given day has a Poisson distribution with parameter $\lambda$ . Use the accompanying data on absences for 50 days to derive a large-sample CI for $\lambda$ . [Hint: The mean and variance of a Poisson variable both equal $\lambda$ , so $Z = ( \bar{X} - \lambda ) / \sqrt { \lambda / n }$ has approximately a standard normal distribution. Now proceed as in the derivation of the interval for p by making a probability statement (with probability 1 - $\alpha$ ) and solving the resulting inequalities for $\lambda$ . \#Absences 0 1 2 3 4 5 6 7 8 9 10 Frequency 1 6 8 10 8 7 5 3 2 1 1

The ability of a confidence interval to contain the value of the population mean $\mu$ is described by the __________.

If a confidence level of 90% is used to construct a confidence interval for the mean $\mu$ of a normal population when the value of the standard deviation $\sigma$ is known, the z critical value is __________.

The formula used to construct a 95% confidence interval for the mean $\mu$ of a normal population when the value of the standard deviation $\sigma$ is known is given by __________.

The 5^th percentile of a chi-squared distribution with 10 degrees of freedom is equal to __________.

A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.50 MPa and a sample standard deviation of .80 MPa. a. Calculate and interpret a 95% lower confidence bound for the true average proportional limit stress of all such joints. What, if any, assumptions did you make about the distribution of proportional limit stress? b. Calculate and interpret a 95% lower prediction bound for the proportional limit stress of a single joint of this type.

A large-sample lower confidence bound for the population mean $\mu \text { is given by } \mu \rangle$ __________.

When $\bar { X }$ is the mean of a random sample of size n (n is large) from a normal population with mean $\mu$ , the random variable $Z = ( \bar { X } - \mu ) / ( S / \sqrt { n } )$ has approximately a __________ distribution with mean value of __________ and standard deviation of __________.

The standard normal random variable $Z = ( \bar { X } - \mu ) / ( \sigma / \sqrt { n } )$ has a mean value of __________ and standard deviation of __________.

Which of the following statements are not true if $t _ { v }$ denotes the density function curve for a t-distribution with $v$ degrees of freedom?

A random sample of 100 observations produced a sample proportion of .25. An approximate 90% confidence interval for the population proportion p is