Exam 7: Statistical Intervals Based on a Single Sample

In developing a confidence interval for the population mean $\mu$ , a sample of 50 observations was used, and the confidence interval was 15.24 $\pm$ 1.20. Had the sample size been 200 instead of 50, the confidence interval would have been

A random sample of 50 observations produced a mean value of 55 and standard deviation of 6.25. The 95% confidence interval for the population mean $\mu$ is between __________ and __________. (two decimal places)

Which of the following statements are not true?

If the random sample $X _ { 1 } , \ldots \ldots , X _ { n }$ is taken from a normal distribution with mean value $\mu$ and standard deviation $\sigma$ , then regardless of the sample size n, the sample mean $\bar { X }$ is distributed with expected value __________ and standard deviation __________.

The area under a t-density curve between the critical values $- t _ { \alpha / 2 , n - 1 } \text { and } t _ { \alpha/ 1 , n - 1 }$ is __________.

The amount of lateral expansion (mils) was determined for a sample of n = 9 pulsed-power gas metal arc welds used in LNG ship containment tanks. The resulting sample standard deviation was s = 2.80 mils. Assuming normality, derive a 95% CI for $\sigma ^ {2 }$ and for $\sigma$ .

A 99% confidence interval for the population mean $\mu$ is determined to be (65.32 to 73.54). If the confidence level is reduced to 90%, the 90% confidence interval for $\mu$

Which of the following statements are not true?

A CI is desired for the true average stray-load loss μ (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with $\mathcal { \sigma }$ = 3.0. a. Compute a 95% CI for μ when n = 25 and $\bar { x }$ = 60. b. Compute a 95% CI for μ when n = 100 and $\bar { x }$ = 60. c. Compute a 99% CI for μ when n = 100 and $\bar { x }$ = 60. d. Compute an 82% CI for μ when n = 100 and $\bar { x }$ = 60. e. How large must n be if the width of the 99% interval for μ is to be 1.0?

A random sample of 100 lightning flashes in a certain region resulted in a sample average radar echo duration of .81 sec and a sample standard deviation of .34 sec. Calculate a 99% (two-sided) confidence interval for the true average echo duration μ , and interpret the resulting interval.

The area under a chi-squared curve with 10 degrees of freedom, which is captured between the two critical values $\hat { \lambda} _ { \alpha / 2 } ^ { 2 } \text { and } \tilde { \lambda } _ { 1 - \alpha / 2 } ^ { 2 }$ is __________.

Which of the following statements are true?

The lower 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:

A more extensive tabulation of t critical values than what appears in your text shows that for the t distribution with 20 df, the areas to the right of the values .687, .860, and 1.064 are .25, .20, and .15, respectively. What is the confidence level for each of the following three confidence intervals for the mean μ of a normal population distribution? Which of the three intervals would you recommend be used, and why? a. $( \bar { x } - .687 s / \sqrt { 21 } , \bar { x } + 1.725 s / \sqrt { 21 } )$ b. $( \bar { x } - .860 s / \sqrt { 21 } , \bar { x } + 1.325 s / \sqrt { 21 } )$ c. $( \bar { x } - 1.064 s / \sqrt { 21 } , \bar { x } + 1.064 s / \sqrt { 21 } )$

The 90^th percentile of a chi-squared distribution with 15 degrees of freedom is equal to __________.

A 99% confidence interval for the mean $\mu$ of a normal population when the standard deviation $\sigma$ is known is found to be 98.6 to 118.4. If the confidence level is reduced to .95, the confidence interval for $\mu$

Which of the following statements are true when $\bar { X }$ is the mean of a random sample of size n from a normal distribution with mean $\mu$ ?

Consider the 1000 95% confidence intervals (CI) for μ that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of μ ? What is the probability that between 950 and 970 of these intervals contain the corresponding value of μ ? (Hint: Let Y = the number among the 1000 intervals that contain μ . What kind of random variable is Y?).

Which of the following statements are true?

When $\bar { X }$ is the mean of a random sample of size n (n is small) from a normal population with mean $\mu$ , the random variable $T = ( \bar { X } - \mu ) / ( S / \sqrt { n } )$ has a probability distribution called t-distribution with n-1 __________.