Chapter 6 and 7.1

Suppose X ~ N (-1,4). What is the z-score of x = 10?

z = 10-(-1)/4 = 2.75 z = 2.75

Suppose X ~ N (11,1). What value of x has a z-score of -2.25?

-2.25 = x-11/1 1(-2.25) = x-11/(1)(1) 11+(-2.25) = x-11+11 8.75 = x

Suppose X ~ N (4,2). What is the z-score of x = 4?

z = 4-4/2 = 0 z = 0

If the area to the left of x in a normal distribution is 0.163, what is the area to the right?

1 - 0.163 = 0.837

A company manufactures rubber balls. The mean diameter of a ball is 12cm with a standard deviation of 0.22cm. Define the random variable X in words.

X = diameter of a rubber ball

About what percent of the x values from a normal distribution lie within one standard deviation (left and right) of the mean of that distribution?

68%

What does z-score measure?

A z-score measures the number of standard deviations a value is from the mean

About what percent of the x values from a normal distribution lie within two standard deviations (left and right) of the mean of that distribution?

95%

About what percent of the x values from a normal distribution lie within three standard deviation (left and right) of the mean of that distribution?

99.7%

About what percent of the x values from a normal distribution lie between the first and third standard deviation on both sides of the mean of that distribution?

31.7% 68% is the first 99.7% is the third 99.7%-68% = 31.7%

Suppose a normal distribution has a mean of 7 and a standard deviation of 1.75. What is the z-score of x = 5.5?

z = 5.5-7/1.75 = -0.86 z = -0.86

Kyle's doctor told him that the z-score for his systolic blood pressure is 1.75. Which of the following is the best interpretation of this standardized score? The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with mean μ = 122 and standard deviation σ = 13. If X = a systolic blood pressure score then X ~ N(122, 13)

Kyle's systolic blood pressure is 1.75 standard deviations above the average systolic blood pressure for men. Calculate Kyle's blood pressure. Mean = 122 Standard Deviation = 13 Z = 1.75 1.75 = x - 122 /13 (13)(1.75) = x - 122 /(13)(13) 22.75 = x - 122 122 + 22.75 = x - 122 + 122 x = 144.75 Kyle's blood pressure

The length of time it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of 4 minutes and a standard deviation of 3 minutes. Seventy percent of the time, it takes more than how many minutes to find a parking space?

Mean = 4 Standard Deviation = 3 70% it takes more than x Use calculator to find invnorm(.30, 4, 3) 2.43

The length of time it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of 5 minutes and a standard deviation of 2 minutes. Find the probability that it takes at least 8 minutes to find a parking space.

Mean = 5 Standard Deviation = 2 To find use calculator normalcdf(8, 10^99, 5, 2) 0.0668

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.1 days and a standard deviation of 2.3 days. What is the 70th percentile for recovery times?

Mean = 5.1 Standard Deviation = 2.3 To find percentiles use calculator 70th percentile is represented 0.70 invnorm(0.70, 5.1, 2.3) = 6.31 70th percentile = 6.31

The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005-2006 season. The heights of basketball players have an approximate normal distribution with mean, μ = 79 inches and a standard deviation, σ = 3.89 inches. For the following height, calculate the z-score and interpret it using complete sentences. 84 inches

Mean = 79 Standard Deviation = 3.89 x = 84 To find z-score: 84-79/3.89 = 1.29 The z-score is 1.29. An NBA player whose height is 84 inches is taller than the average. This is because 84 is more than the mean.

According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let X = height of the individual. The middle 40% of heights fall between what two values? Write the probability statement.

P(x1 < X < x2) = .40 To find middle 40% values first subtract 1 - .40 = .60 Divide that by 2 so we get the percentage from each side =.30 30th percentile finds x1 70th percentile finds x2 (because .30 + .40) x1 = invnorm(.30, 66, 2.5) = 64.68899872 x2 = invnorm(.70, 66, 2.5) = 67.31100128 x1 = 64.7 x2 = 67.3

For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. If one person took the SAT math test and scored 700 and a second person took the ACT math test and scored 31, who did better with respect to the test they took?

SAT Mean - 514 Standard Deviations = 117 x = 700 z = 700-514 /117 z = 1.59 ACT Mean = 21 Standard Deviations = 5.3 x = 31 z = 31-21 /5.3 z = 1.89 ACT did better because higher z-score

Use the following info to answer the next exercise X ~ N (54,8) Find the 80th percentile

Solve with calculator invnorm(.80, 54, 8) = 60.73

Use the following info to answer the next exercise X ~ N (54,8) Find the probability that x > 59

Solve with calculator normalcdf(59, 10^99, 54, 8) = 0.2660

The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005-2006 season. The heights of basketball players have an approximate normal distribution with mean, μ = 79 inches and a standard deviation, σ = 3.89 inches. If an NBA player reported his height had a z-score of 3.5, would you believe him?

We know the z-score so we need to write out the equation then rearrange to solve for x. z = 3.5 3.5 = x-79/3.89 (3.89)(3.5) = x-79/(3.89)(3.89) 13.615 = x - 79 79 + 13.615 = x - 79 + 79 92.615 = x A z-score of 3.5 equates to a height of 92.615 inches. There are few NBA players this tall, so it is unlikely that the players z-score is 3.5

In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean μ = 520 and standard deviation σ = 115. What math SAT score is 1.5 standard deviations above the mean? What can you say about this SAT score?

Mean = 520 Standard Deviation = 115 (115)(1.5) = 172.5 172.5 + 520 = 692.5 692.5 is 1.5 standard deviations above the mean.

In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean μ = 520 and standard deviation σ = 115. Calculate the z-score for an SAT score of 710. Interpret it using a complete sentence.

Mean = 520 Standard Deviation = 115 z = 710 Z = 710 - 520 /115 = 1.65 The exam score of 710 is 1.65 standard deviations above the mean of 520.

According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let X = height of the individual. Would you expect to meet many Asian adult males over 73 inches? Explain why or why not, and justify your answer numerically.

Mean = 66 Standard Deviation = 2.5 To find probability of being over 73 inches, use calculator normalcdf(73, 10^99, 66, 2.5) No, because the probability that an Asian male is over 73 inches tall is 0.0026

The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005-2006 season. The heights of basketball players have an approximate normal distribution with mean, μ = 79 inches and a standard deviation, σ = 3.89 inches. For the following height, calculate the z-score and interpret it using complete sentences. 76 inches

Mean = 79 Standard Deviation = 3.89 x = 76 To find z-score: 76-79/3.89 = -0.77 The z-score is -0.77 An NBA player whose height is 76 inches is shorter than average. This is because 76 is less than the mean