Exam 6: The Normal Distribution

Find the indicated probability or percentage for the normally distributed variable. -A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a Rating that is between 200 and 275.

Provide an appropriate response. -The area under the standard normal curve between 1 and 2 is equal to 0.1359. Scores on a particular aptitude test are normally distributed with a mean of 100 and a standard deviation of 10. Which of The following are equal to 13.59%? a. The percentage of scores between 120 and 130 b. The percentage of scores between 110 and 120 c. The percentage of scores between 80 and 90 d. The percentage of scores between 90 and 120 e. The percentage of scores between 90 and 110

Find the indicated probability or percentage for the normally distributed variable. -A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a Rating that is between 170 and 220.

Use a table of areas for the standard normal curve to find the required z-score. -Find the z-score for which the area under the standard normal curve to its left is 0.04

Construct a normal probability plot of the given data. -The final exam scores for 15 students in a statistics course are given below. 95 70 60 82 40 62 97 57 32 75 78 85 68 54 88

Use a table of areas for the standard normal curve to find the required z-score. -Find the z-score having area 0.09 to its left under the standard normal curve.

Provide an appropriate response. Assume that the variable under consideration has a density curve. -Given that 40.5% of all possible observations of the variable are less than 12, determine the area under the density curve that lies to the right of 12.

Provide an appropriate response. -A curve has area 0.296 to the left of 5 and area 0.642 to the right of 5. Could this curve be a density curve for some variable? Explain your answer.

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, exactly 225 are defective.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -Estimate the probability that in 200 rolls of a balanced die, the number of sixes is between 36 and 45 inclusive.

Use a table of areas to find the specified area under the standard normal curve. -The area that lies between 0 and 3.01

Use a table of areas to obtain the shaded area under the standard normal curve. -

Use a table of areas to obtain the shaded area under the standard normal curve. -

Find the specified percentile, quartile, or decile. -The weights of certain machine components are normally distributed with a mean of 8.98 g and a standard deviation of 0.05 g. Find the 97th percentile.

Solve the problem. -A northeastern college has an enrollment of 2835 female students. Records show that the mean height of these students is $65.2$ inches and that the standard deviation is $2.8$ inches. The table shows frequency and relative-frequency data for these heights. If you assume that the distribution of heights is approximately normal, then you can use the table to estimate areas under the associated normal curve (that is, under the normal curve that has parameters $\mu = 65.2$ and $\sigma = 2.8$ ). Making this assumption, estimate the area under the associated normal curve between 65 and 70 . Height (inches) Freq. Relative freq. 57-under 58 16 0.0056 58 -under 59 25 0.0088 59 -under 60 56 0.0198 60-under 61 96 0.0339 61-under 62 178 0.0628 62-under 63 263 0.0928 63-under 64 322 0.1136 64-under 65 388 0.1369 65-under 66 406 0.1432 66-under 67 361 0.1273 67-under 68 292 0.1030 68-under 69 201 0.0709 69-under 70 113 0.0399 70-under 71 75 0.0265 71-under 72 25 0.0088 72-under 73 16 0.0056 73-under 74 2 0.0007

Fill in the blanks by standardizing the normally distributed variable. -Dave drives to work each morning at about the same time. His commute time is normally distributed with a mean of 51 minutes and a standard deviation of 5 minutes. The percentage of Time that his commute time exceeds 58 minutes is equal to the area under the standard normal Curve that lies to the ___ of ___.

Provide an appropriate response. -A variable is normally distributed with a mean of 100 and a standard deviation of 10. A student wanted to find the percentage of observations of the variable lying between 106 and 110. What is wrong with his solution? $\underline { \text { Student's solution: } }$ $\mathrm { z }$ -scores: $\frac { 106 - 100 } { 10 } = 0.6 \quad \frac { 110 - 100 } { 10 } = 1.0$ Percentage of scores lying between 106 and 110 = difference between z -scores =1.0-0.6=0.4=40\%

A normal probability plot is given below for a sample of scores on an aptitude test. Use the plot to identify outliers, if any. Explain your reasoning.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -With n = 20 and p = 0.60, estimate P(less than or equal to 8).

Solve the problem. -Frequency data were reported for the ages of women who became mothers during one year in a selected U.S. city. The age distribution is given in the table. Obtain a relative-frequency histogram Of these data and determine whether the ages are approximately normally distributed. Age (yrs) Frequency 10-under 15 10 15-under 20 87 20-under 25 888 25-under 30 679 30 -under 35 245 35 -under 40 227 40 -under 45 123 45 -under 50 79