Exam 6: The Normal Distribution

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 right of 62. 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

A normal probability plot is given below for the weekly incomes (in dollars)of a sample of engineers in one town. Use the plot to identify outliers, if any. Explain your reasoning.

Fill in the blanks. A fair coin is flipped 280 times. You wish to find the probability that the number of tails is greater than 160 . This probability can be estimated by finding the area to the right of ______under the normal curve with $\mu =$ _____ and $\sigma =$ _____

Find the specified percentile, quartile, or decile. -Suppose that replacement times for washing machines are normally distributed with a mean of 11 years and a standard deviation of 1 years. Find the 82nd percentile.

Suppose that you know the area under the standard normal curve to the right of 2 and the area under the standard normal curve to the right of 1. Without further consulting a table of areas, how could you find the area under the standard normal curve between 1 and 2? Explain your reasoning.

Find the indicated probability or percentage for the normally distributed variable. -The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected Bottle will be less than 32 oz?

Provide an appropriate response. -Which of the following statements concerning the standard normal curve is/are true (if any)? a. The area under the standard normal curve to the left of -3 is zero. B. The area under the standard normal curve between any two z-scores is greater than zero. C. The area under the standard normal curve between two z-scores will be negative if both z-scores Are negative. D. The area under the standard normal curve to the left of any z-score is less than 1.

Fill in the blanks by standardizing the normally distributed variable. -The amount of time that customers wait in line during peak hours at one bank is normally distributed with a mean of 13 minutes and a standard deviation of 3 minutes. The percentage of Time that the waiting time lies between 11 and 13 minutes is equal to the area under the standard Normal curve between ___ and ___.

Use the empirical rule to solve the problem. -The annual precipitation for one city is normally distributed with a mean of 340 inches and a standard deviation of 3.8 inches. Fill in the blanks. In 95.44% of the years, the precipitation in this city is between ___ and ___ inches.

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 25 -under 30 23 30 -under 35 25 35 -under 40 24 40 -under 45 30 45 -under 50 26 50 -under 55 22 55 -under 60 27 60 -under 65 30 65 -under 70 19

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 35 minutes and a standard deviation of 5 minutes. The percentage of Time that his commute time is less than 44 minutes is equal to the area under the standard normal Curve that lies to the ___ of __. right,

Explain why a continuity correction factor is necessary when approximating the binomial distribution by the normal distribution. Refer to the terms "discrete" and "continuous", and draw a diagram to support your answer.

Provide an appropriate response. -Which of the following statements concerning areas under the standard normal curve is/are true? a. If a $z$ -score is negative, the area to its right is greater than $0.5$ b. If the area to the right of a $z$ -score is less than $0.5$ , the $z$ -score is negative. c. If a $\mathrm { z }$ -score is positive, the area to its left is less than $0.5$

Use the empirical rule to solve the problem. -The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 210 hours and a standard deviation of 5 hours. What percentage of the bulbs have lifetimes that lie within 1 Standard deviation to either side of the mean?

Find the specified percentile, quartile, or decile. -The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 263 hours and a standard deviation of 8 hours. Find the first quartile, Q1 .

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

Find the specified percentile, quartile, or decile. -The annual precipitation for one city is normally distributed with a mean of 26 inches and a standard deviation of 4 inches. Find the 2nd decile.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -In one county, the conviction rate for speeding is 85%. Estimate the probability that of the next 100 speeding summonses issued, there will be at least 90 convictions.

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.

Generally, the normal probability plot for a data set must be roughly linear in order to assume that the variable is approximately normally distributed. Should this rule be interpreted more strictly for small data sets or for large data sets? Explain.