Exam 6: The Normal Distribution

Solve the problem. -Data were reported for household size (number of people in household)in a small community. The size distribution is given in the table. Obtain a relative-frequency histogram for these data and Determine whether the household sizes are approximately normally distributed. Size (\# of people) Frequency 1-under 2 4 2-under 3 33 3-under 4 30 4-under 5 12 5-under 6 6 6-under 7 3

Use a table of areas for the standard normal curve to find the required z-score. -Find the z-score for having area 0.09 to its right under the standard normal curve, that is, find ${ } ^ { \mathrm { z } } 0.09$

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

Scores on an aptitude test are normally distributed with a mean of 400 and a standard deviation of 60. Explain how you would find any given percentile.

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

A normal probability plot is given below for the lifetimes (in hours)of a sample of batteries of a particular brand. Use the plot to assess the normality of the lifetimes of these batteries. Explain your reasoning.

Suppose that you know the area under the standard normal curve between 1 and 3 and the area under the standard normal curve to the left of 3. Without further consulting a table of areas, how could you find the area under the standard normal curve to the left of 1? Explain your reasoning by using a sketch of the standard normal curve.

Frequency data were reported for the ages of full-time employees at a company. 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 25 -under 30 20 30 -under 35 65 35 -under 40 60 40 -under 45 112 45 -under 50 266 50 -under 55 300 55 -under 60 385 60 -under 65 200 65 -under 70 36

Find the specified percentile, quartile, or decile. -The serum cholesterol levels for men in one age group are normally distributed with a mean of 178.2 and a standard deviation of 40.8. All units are in mg/100 mL. Find the 9th percentile.

Find the specified percentile, quartile, or decile. -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. Find the 6th decile.

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

Construct a normal probability plot of the given data. -The weekly incomes (in dollars)of a sample of 12 nurses working at a Los Angeles hospital are given below. 500 750 630 480 550 650 720 780 $820 \quad 960 \quad 1200 \quad 770$

The mean of a normally distributed variable can be any real number.

A student wished to use a table of areas for the standard normal curve to find the z-score having an area to its right of 0.52. The student started by looking for the closest area to 0.52 in the body of the table and reading off the corresponding z-score which was 0.05. She then subtracted this z-score from 1 to get 0.95. Was her reasoning correct? If not, where did she go wrong and how would you have solved the problem?

Find the indicated probability or percentage for the normally distributed variable. -The incomes of trainees at a local mill are normally distributed with a mean of $1,100 and a standard deviation $150. What percentage of trainees earn less than $900 a month?

Use a sketch of the standard normal curve to explain the difference between z-scores and areas under the standard normal curve. What are the possible values for an area and what are the possible values for a z-score?

Provide an appropriate response. -Most problems involving normally distributed variables are one of two types. Type A: Find a probability or percentage, e.g. find the probability that X lies in a specified range. Type B: Find the observation corresponding to a given probability or percentage. Suppose that scores on a test are normally distributed with a mean of 80 and a standard deviation Of 8. Which of the following questions below are of type B? A. Find the 80th percentile. B. Find the cutoff for the A grade if the top 10% get an A. C. Find the percentage scoring more than 90. D. Find the score that separates the bottom 30% from the top 70%. E. Find the probability that a randomly selected student will score more than 80.

Choose the one alternative that best completes the statement or answers the question.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 0.40

Provide an appropriate response. -Fill in the blanks. A balanced die is rolled 210 times. You wish to find the probability that the number of fives is between 30 and 38 inclusive. This probability can be estimated by finding the area between____ and _______under the normal curve with $\mu =$ ___- and $\sigma =$ ____

Solve the problem. -A southeastern college has an enrollment of 2951 female students. Records show that the mean height of these students is 64.7 inches and that the standard deviation is 2.3 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 µ = 64.7 and Ϭ = 2.3). Making This assumption, estimate the area under the associated normal curve to the left of 65. Height (inches) Freq. Relative freq. 56-under 57 2 0.0007 57-under 58 7 0.0024 58-under 59 14 0.0047 59-under 60 31 0.0105 60-under 61 101 0.0342 61-under 62 194 0.0657 62-under 63 311 0.1054 63-under 64 410 0.1389 64-under 65 526 0.1782 65-under 66 482 0.1633 66-under 67 397 0.1345 67-under 68 254 0.0861 68-under 69 150 0.0508 69-under 70 49 0.0166 70-under 71 17 0.0058 71-under 72 5 0.0017 72-under 73 1 0.0003