Exam 6: The Normal Distribution

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -Estimate the probability of getting exactly 43 boys in 90 births.

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

Provide an appropriate response. Assume that the variable under consideration has a density curve. -The area under the density curve that lies between 26 and 48 is 0.409. What percentage of all possible observations of the variable are either less than 26 or greater than 48?

Solve the problem. -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 20-under 25 5 25-under 30 60 30-under 35 443 35-under 40 340 40 -under 45 231 45-under 50 75 50 -under 55 87 55 -under 60 35 60-under 65 15

Does the presence of an outlier in your data set necessarily mean that you cannot use the normal model to interpret the data? Explain.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses, estimate the probability of getting at least 20% correct.

Provide an appropriate response. -Fill in the blanks. A balanced die is rolled 240 times. You wish to find the probability that the number of ones is at most 50 . This probability can be estimated by finding the area that is to the left of________ under the normal curve with $\mu =$ ________and $\sigma =$ __________

Find the indicated probability or percentage for the normally distributed variable. -The variable $X$ is normally distributed. The mean is $\mu = 60.0$ and the standard deviation is $\sigma = 4.0$ . Find $P ( X < 53.0 )$ .

Suppose that you know the area under the standard normal curve to the right of -1.7. How could you use this to find the area under the standard normal curve to the right of 1.7? Explain your reasoning.

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

Use a table of areas for the standard normal curve to find the required z-score. -Find the z-score having area 0.86 to its right under the standard normal curve; that is, find ${ } ^ { Z } 0.86 ^ { . }$

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -The probability that a radish seed will germinate is 0.7. Estimate the probability that of 140 randomly selected seeds, exactly 100 will germinate.

Find the indicated probability or percentage for the normally distributed variable. -Assume that the weights of quarters are normally distributed with a mean of 5.67 g and a standard deviation 0.070 g. A vending machine will only accept coins weighing between 5.48 g and 5.82 g. What percentage of legal quarters will be rejected?

Use a table of areas for the standard normal curve to find the required z-score. -Find $\mathrm { z } _ { 0 } 0.45 ^ { . }$

Fill in the blanks. A fair coin is flipped 360 times. You wish to find the probability that the number of tails is less than 150 . This probability can be estimated by finding the area that is to the left of ______under the normal curve with $\mu =$ ______and $\sigma =$ ______

Construct a normal probability plot of the given data. -The resting heart rates from a group of 9 men before starting a workout program are given below. 58 70 63 48 60 55 45 51 69

A normal probability plot is given below for a sample of scores on an aptitude test. Use the plot to assess the normality of scores on this test. Explain your reasoning.

Use a table of areas to find the specified area under the standard normal curve. -The area that lies between -1.10 and -0.36

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 16 15 -under 20 134 20-under 25 534 25 -under 30 1043 30 -under 35 982 35 -under 40 399 40 -under 45 94 45-under 50 7

Suppose that scores on a test are normally distributed with a mean of 80 and a standard deviation of 8. You have been asked to find the 70th percentile. After sketching a standard normal curve and shading the area of interest, the next step in solving this problem is to use the table of areas. Would you look for 0.7 in the body of the table or in the left-hand column? Explain your reasoning.