Chapter 11: The Normal Distributions

A good use of z-scores is to compare values in two different distributions. Suppose pieces of chocolate pie served at a restaurant average 350 calories with a standard deviation of 20 calories and pieces of apple pie have an average of 425 calories with a standard deviation of 30 calories. What is the z-score for a piece of apple pie with 405 calories? Answer to two decimal places.

-0.67 The z-score for the apple pie is (405 - 425)/30 = -0.67

For a Normal distribution, the chance of the variable z equaling exactly 1.5 is _____.

0 For a standard Normal (or any Normal distribution) the chance of the variable exactly equaling any value is 0.

According to the 68-95-99.7 Rule, approximately 99.7% of the area in any Normal distribution is within 3 standard deviations of the mean μ. The actual number z of standard deviations for 99.7% of all observations within z σ of μ is 2.97. This is different from the Empirical Rule by _____ standard deviations.

0.03 If 99.7% is in the center of the distribution, there is 0.0015 (0.15%) in either tail. Read the Standard Normal table to find that z = 2.97. This is a difference of 3 - 2.97 = 0.03 from the Empirical Rule

Use the standard normal table (use the table reader, the table in your textbook, or a graphing calculator) to find the area between the following two z-scores: -1.99 and 0.50. Answer to four decimal places

0.6682 The area to the left of 0.50 is 0.6915. The area to the left of -1.99 is 0.0233. For the area between the two, subtract 0.6915 - 0.0233 = 0.6682

IQs as measured by the Wechsler Adult Intelligence Scale are approximately Normal with mean 100 and standard deviation 15. About 84% of people will have IQs below 115 because 115 is _____ standard deviation(s) above the mean.

1 115 is one standard deviation above the mean. We expect about 68% of all IQs within one standard deviation of 100. The remaining 32% is split in half because of symmetry. Add the lower 16% to the 68% in the middle of the distribution.

Using the standard Normal table (use Table A in your textbook, or a graphing calculator), the area to the right of z = 0.94 is _____ - 0.8264 = 0.1736.

1 The table (or a graphing calculator) gives the area to the left of 0.94 as 0.8264. The area to the right will be 1 - 0.8264 = 0.1736. Note that a z-score of -0.94 would also give the same result because the graph is symmetric.

Mensa is a society for "geniuses." One way to qualify for membership is having an IQ at least 2.5 standard deviations above average. IQs are approximately Normally distributed, according to the WAIS scale. What percent of people should qualify? The table gives you 0.9938, so 1 - 0.9938 = 0.0062; therefore, 0.62% of people should qualify. The table gives you 0.938, so 1 - 0.938 = 0.62; therefore, 6.2% of people should qualify. The table gives you 0.5987, therefore 59.87% of people should qualify The table gives you 0.9938, therefore 99.38% of people should qualify

The table gives you 0.9938, so 1 - 0.9938 = 0.0062; therefore, 0.62% of people should qualify To qualify, a person's z-score must be at least 2.5. Find the area above z = 2.5 by subtracting the table entry (0.9938) from 1.

A Normal quantile plot is used to determine if data is __________ distributed.

Normally A Normal quantile plot is used to test if data is normally distributed. The plot will show a straight line pattern if the data is Normally distributed.

How does finding a value in a distribution with a specified percent either above or below it compare to the process of finding the relative proportion of observations less than the value? One process is the reverse of the other The process with percents has an extra step at the end that the proportion process does not have The two processes are the same The two processes have no similarities

One process is the reverse of the other When we know the proportion of the values less than the value of interest, we locate the proportion in the body of the table and read left and up to find the corresponding z-score. This is the reverse of the process used when we know the z-score and want to find proportions.

Mensa is the "high IQ" society. Their rules for eligibility for membership state that an individual must have an IQ in the upper 2%, which corresponds to a z-score of 2.05. The Wechsler Adult Intelligence Scale is approximately Normal with mean 100 and standard deviation 15. On this scale, an IQ _____ 130 will qualify. (Mensa does not use decimal points in their eligibility scores.) greater than greater than or equal to less than less than or equal to

greater than 100 + 2.05(15) = 130.75, so 130 is too low to be a qualifying member of Mensa

If a z-score is negative, the area to its right is __________ 0.5. greater than less than equal to approximately

greater than If a z-score is negative, the value is below the mean. The area to the right will be more than 0.5

Aldo takes a nationally administered college readiness test and finds out his z-score was 1.7. Does this mean Aldo's score was lower or higher than the national average? His score was __________ than the national average.

higher Aldo's z-score was 1.7, this is a positive number, so Aldo's score was higher than the national average

If the data in a data set are not Normally distributed, the points in the normal quantile plot will show a pattern that is not __________.

linear The pattern will only be linear if the data are Normally distributed.

John takes a nationally administered college readiness test and finds out his z-score was -2.1. Does this mean John's score was lower or higher than the national average? His score was __________ than the national average

lower John's z-score was -2.1, this is a negative number, so John's score was less than the national average

In a Normal quantile plot, points that are far away from the overall pattern of the plot represent __________

outliers Outliers will appear as points that are far away from the overall pattern of the other points

If the area to the right of a z-score is less than 0.5, the z-score is __________. 0 less than 0.5 negative positive

positive If the area to the right is less than 0.5, the value is above the mean so the z-score is positive

A Normal __________ plot allows us to test if data is Normally distributed by looking for a straight line pattern in the plot.

quantile A normal quantile plot is a useful tool for assessing if data is normally distributed. It can provide more information than might be visible in a histogram or stemplot of the data.

You will find the __________ of any Normal distribution 0.675 standard deviations from the mean.

quartiles 0.7500 is located in the table between 0.7486 and 0.7517. These correspond to z = 0.67 and 0.68. So, there is 75% of the area below z = 0.675. That makes it Q3. Similarly, there is 25% of the area below z = -0.675

Normal density curves are: bell-shaped single-peaked symmetric symmetric, single-peaked, and bell-shaped

symmetric, single-peaked, and bell-shaped Normal density curves are symmetric, bell-shaped, and single-peaked

The heights of women aged 30 to 39 are approximately Normal with mean 63 inches and standard deviation 3 inches. Eva and Marita are both 35 years old. The z-score for Eva's height is 0.7 and the z-score for Marita's height is 1.6. Marita is ___________ Eva. taller than the same height as shorter than not enough information to determine who is taller

taller than The z-score for Marita's height is larger than the z-score for Eva's height, so Marita must be taller than Eva

Normal distributions are completely specified by the mean, μ. x¯ and s. the standard deviation, σ. μ and σ.

μ and σ. It takes both of these to completely specify a Normal distribution.

In a certain marathon, the average time to complete the race was Normally distributed with mean μ = 4.15 hours and standard deviation σ = 0.84 hours. If we wanted to give medals to the 4% of the runners with the best times, what time in hours would be the slowest time to get a medal? Note: The "best" times are the smallest times, we are looking for the cut-off score for the bottom 4% of the distribution of running times. Therefore, you start by locating _____ in the z-score table to find the accompanying z-score. This leads to the "best" times of 2.68 hours. Round your answer to two decimal places.

0.04 Locate 0.0400 as closely as possible in the body of the table (-1.75 has area 0.0401 to the left of it). Read to the left for -1.7 and up for .05. Now, use X = zσ + μ to solve for X = 2.68 hours. When you substitute -1.75 into the equation you have -1.75*0.84 + 4.15 = 2.68

Hemoglobin is the compound in red blood cells that carries oxygen to the body. The distribution of hemoglobin in women in g/dl of blood is approximately normally distributed with mean 14 and standard deviation 1. Too little hemoglobin (below 12), and you're anemic. Too much (above 15), and (unless you live at high altitudes), you can have other problems. Using the 68-95-99.7% rule, 2.5% of women will have hemoglobin levels below 12 (be anemic) because __________. 12 is two standard deviations below the mean 12 is one standard deviation below the mean 12 is three standard deviations below the mean 12 is the mean

12 is two standard deviations below the mean 12 is 2 standard deviations below the mean. Normal distributions have 95% of their area within two standard deviations of the mean, so there is 5% total below 12 and above 16. Because of symmetry, there will be 2.5% below 12.

A university wants to select individuals for their honors program partly on the basis of SAT scores. Total scores for the three parts of the test are approximately normally distributed with mean 1500 and standard deviation 250. If they want only the top 5% to qualify, what total SAT score must they equal or exceed? Remember, SAT scores are multiples of 10. For this reason, please round your answer to the nearest multiple of 10. 1910 because 1.645*250 + 1500 = 1911.25. This rounds to 1910. 1920 because 1.645*250 + 1500 = 1911.25. This means 1910 is too low, so 1920. 1911.25 because 1.645*250 + 1500 = 1911.25. 1900 because 1.645*250 + 1500 = 1911.25 and that rounds to 1900 as the nearest hundredth.

1920 because 1.645*250 + 1500 = 1911.25. This means 1910 is too low, so 1920. The z-score with 95% of area under the curve to its left is 1.645. SAT = 1.645*250 + 1500 = 1911.25, so they must score at least 1920 in multiples of 10.

The tallest person ever confirmed by the Guinness records organization was Robert Wadlow of the United States, at 8 feet 11.1 inches tall. Heights of American men are approximately Normal with mean 69.3 inches and standard deviation 2.8 inches. Wadlow's z-score is 13.5 because (107.1 - _____)/2.8 = 13.5.

69.3 Converted to inches, he was 8*12+11.1 = 107.1 inches tall. His z-score is (107.1 - 69.3)/2.8 = 13.5

If data is Normally distributed the normal quantile plot will show: no pattern. an S shaped pattern. a bell-shaped curve. a straight line.

a straight line. Normally distributed data will show a straight line pattern in a Normal quantile plot