Exam 6: Normal Probability Distributions

Solve the problem. Round to the nearest tenth unless indicated otherwise. -The weights of certain machine components are normally distributed with a mean of 8.01 g and a standard deviation of 0.06 g. Find the two weights that separate the top 3% and the bottom 3%. These weights could serve As limits used to identify which components should be rejected. Round to the nearest hundredth of a gram.

Suppose that you wish to find $P ( - 2 < x < 2 )$ ) for a continuous uniform distribution having a minimum of -3 and a maximum of 3. If you incorrectly assume that the distribution is normal instead of uniform, will your answer be too big, too small, or will you still obtain the correct answer? Explain your thinking.

Solve the problem. -A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, Find the probability that their mean rebuild time exceeds 8.1 hours.

Solve the problem. -The number of books sold over the course of the four-day book fair were 194, 197, 247, and 76. Assume that samples of size 2 are randomly selected with replacement from this population of four values. List the different possible samples, and find the mean of each of them.

Assume that the red blood cell counts of women are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells per microliter. Find the 80 $80 ^ { \text {th } }$ percentile for the red Blood cell counts of women.

Solve the problem. -The systolic blood pressures of the patients at a hospital are normally distributed with a mean of 138 mm Hg and a standard deviation of 13.5 mm Hg. Find the two blood pressures having these properties: the mean is Midway between them and 90% of all blood pressures are between them.

Solve the problem. -A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, Find the probability that their mean rebuild time is less than 7.6 hours.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -A certain question on a test is answered correctly by 22% of the respondents. Estimate the probability that among the next 150 responses there will be at most 40 correct answers.

Solve the problem. -A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is greater than 71.

Solve the problem. -Scores on a test have a mean of 70 and $\mathrm { Q } _ { 3 }$ is 83. The scores have a distribution that is approximately normal. Find P90. (You will need to first find the standard deviation.)

For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg. If 23 women aged 18-24 are randomly selected, find the probability that Their mean systolic blood pressure is between 119 and 122 mm Hg.

Find the indicated probability. -The incomes of trainees at a local mill are normally distributed with a mean of $1100 and a standard deviation of $150. What percentage of trainees earn less than $900 a month?

Solve the problem. -When sampling without replacement from a finite population of size N, the following formula is used to find the standard deviation of the population of sample means: $\sigma _ { \mathrm { x } } = \frac { \sigma } { \sqrt { \mathrm { n } } } \sqrt { \frac { \mathrm { N } - \mathrm { n } } { \mathrm { N } - 1 } }$ However, when the sample size n, is smaller than 5% of the population size, N, the finite population correction factor, $\sqrt { \frac { N - n } { N - 1 } }$ , can be omitted. Explain in your own words why this is reasonable. For N = 200, find the values of the finite population correction factor when the sample size is 10%, 5%, 3%, 1% of the population, respectively. What do you notice?

Solve the problem. -Flood insurance policies sold in the last three days by a new agent were 3, 5, and 6. Assume that samples of size 2 are randomly selected with replacement from this population of three values. List the different possible samples, and find the mean of each of them.

Solve the problem. -For a standard normal distribution, find the percentage of data that are between 3 standard deviations below the mean and 1 standard deviation above the mean.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -Two percent of hair dryers produced in a certain plant are defective. Estimate the probability that of 10,000 randomly selected hair dryers, at least 219 are defective.

Solve the problem. -The probability of exactly 44 green marbles

Solve the problem. -A normal quartile plot is given below for the weekly incomes (in dollars) of a sample of engineers in one town. Use the plot to assess the normality of the incomes of engineers in this town. Explain your reasoning.

An unbiased estimator is a statistic that targets the value of the of the population parameter such that the sampling distribution of the statistic has a ________ equal to the ________ of the corresponding parameter.

Solve the problem. -The scores on a certain test are normally distributed with a mean score of 60 and a standard deviation of 5. What is the probability that a sample of 90 students will have a mean score of at least 60.527?