Unit 2.6: Corequisite
(C) Robert's Z-score =
-0.35. Z = (70 - 71.1) / 3.1 = -0.35
An apple in the batch has a rating of 60. Find the Z-score of this rating. Round the Z-score to two decimal places
-0.98
9) The table below displays the name, sex, and height of two college students. We saw earlier that male college students have a mean height of μ = 71.1 inches with a standard deviation of σ = 3.1 inches, and female college students have a mean height of μ = 65.2 inches with a standard deviation of σ = 2.9 inches. Find the Z-score of each student's height. Round each Z-score to two decimal places. Interpret each Z-score. (A) Jada's Z-score =
0.62. Z = (67 - 65.2) / 2.9 = 0.62
(7) Suppose that x represents a random male college student's height. Find: (A) P(74.2 ≤ x ≤ 80.4)
15.85% 0.135 + 0.0235 = 0.1585 = 15.85%
(B) P(x ≤ 62.3)
16% 0.0015 + 0.0235 + 0.135 = 0.16 = 16%
(10) Another female student, Marcia, has a height with a Z-score of 1.75. What is Marcia's height in inches? Round to one decimal place.
70.3 We can find her height in two ways. (1) Her height is 1.75 standard deviations above the mean, so we can add (1.75)(2.9) to 65.2, to get: 65.2 + (1.75)(2.9) = 70.3. (2) In the Z-score formula, we can substitute 1.75 for Z, 65.2 in for μ, and 2.9 in for σ, and then solve the equation for x. 1.75 = (x - 65.2) / 2.9. The solution is x = 70.3 inches. 70.3 inches corresponds to a height of about 5 feet 10 inches.
An apple in the batch has a Z-score of 0.6. What is the rating of this apple? Round to the nearest whole number.
73 Solve the equation for x: 0.6 = (x - 68) / 8.2. The solution is x = 72.92, so the apple has a rating of about 73.
(5) Suppose that x represents a random female college student's height. Use the Empirical Rule to find: (A) P(59.4 ≤ x ≤ 68.1)
81.5% 0.135 + 0.34 + 0.34 = 0.815 = 81.5%
(B) P(x ≥ 64.9)
97.5% The percentage of heights between 64.9 and 77.3 is 0.95. Adding 0.0235 and 0.0015 results in a total probability of 0.975 = 97.5%
1) If we want to assess whether a male college student's height is unusually large, would it make sense to compare the student's height to the population of all college students' heights, or to the population of all male college students' heights? Explain.
Based on these normal distributions, male college students' heights tend to be greater than female college students' heights. Since the populations are different, if we want to assess whether a male student's height is unusual, we should compare it to the population of male students' heights. i wrote: It would be better to compare the student's height with the population of all male college students heights because females vary too much genetically when it comes to height and body proportions. Having females would skew the data
What interval contains the middle 95% of apple ratings?
Between 51.6 and 84.4
What interval contains the middle 95% of orange ratings?
Between 8.4 and 31.6
(4) What heights are in the middle 95% of the distribution shown above? What heights of U.S. female college students are unusual?
Female college students' heights between 59.4 inches and 71 inches are in the middle 95% of the distribution. Heights less than 59.4 inches and greater than 71 inches are considered unusual.
(3) Interpret the standard deviation of each probability distribution.
For female college students, the typical deviation from the mean is 2.9 inches. So, female college students have heights that, on average, deviate from the mean height by 2.9 inches. For male college students, the typical deviation from the mean is 3.1 inches. So, male college students have heights that, on average, deviate from the mean height by 3.1 inches.
(B) Interpret Jada's Z-score.
Jada's height is 0.62 standard deviations above the mean.
What apple ratings should be considered unusual?
Less than 51.6 and more than 84.4
What orange ratings should be considered unusual?
Less than 8.4 and more than 31.6
(11) We can use Z-scores to compare values across different distributions. What is more unusual: a male college student with a height of 6 feet 5 inches or a female college student with a height of 6 feet? Explain.
Male college student height of 6 feet 5 inches is the same as 77 inches. Z = (77 - 71.1) / 3.1 = 1.90. Female college student height of 6 feet is the same as 72 inches. Z = (72 - 65.2) / 2.9 = 2.34. The female height of 6 feet is more unusual.
(6) What heights are in the middle 95% of the distribution shown above? What heights of U.S. male college students are unusual?
Male college students' heights between 64.9 inches and 77.3 inches are in the middle 95% of the distribution. Heights less than 64.9 inches and greater than 77.3 inches are considered unusual.
8) Let's now address the original question: Is your height normal? Use the probability distributions above to answer this question.
No, my height is considered unusual
An orange in the batch has a rating of 30. Is this rating unusual?
No, the Z-score is 1.72, so it is not unusual.
(D) Interpret Robert's Z-score.
Robert's height is 0.35 standard deviations below the mean.
A grocery store owner has an interesting system for rating the quality of her produce. She rates the quality of her apples using a 100 point scale, scoring apple quality from 0 to 100. After rating a large batch of apples she found that the mean rating of the apples is μ = 68 and the population standard deviation for the ratings is σ = 8.2. She rates the quality of her oranges on a 40-point, scoring orange quality from 0 to 40. After rating a large batch of oranges, she found that the mean rating of the oranges is μ = 20 and the population standard deviation for the ratings is σ = 5.8.
The normal curves below show the probability distributions for the ratings of each type of fruit. The mean of each distribution is shown in bold.
Let's compare apples to oranges. Which fruit has the better rating with respect to its distribution: an apple with a rating of 90 or an orange with a rating of 36?
The orange, since it's Z-score is higher.
2) How are the female and male probability distributions similar? How are they different?
The probability distributions have the same shape, but they have different centers and different spreads. The male college students' heights have a greater mean and a greater standard deviation. So, male heights are bigger, on average, then female heights, and male heights have greater variability than female heights. i wrote: If both were compared with each other there would be some similarities. Outliers of each group could have in common with each other. For example, a tall female could range a more typical height of a male and a short male could be similar to a typical height of a female student. The distributions would show some similarities. However, I think the overall distribution would very a lot.
Standard deviations allow us to compare how unusual values from different populations are. Z-scores are used to standardize values. The Empirical Rule applies to only bell shaped, symmetric distributions.
Values that are more than 2 standard deviations from the mean are unusual. The mean of a normal distribution is the center of the distribution.
The Z-score is a standardized measure for the relative position of a particular data value against the values in the population the Z-score of a value, x, is the number of standard deviations that x is from the mean.
When Z is negative, then x is below the mean. When Z is positive, then x is above the mean. When Z is zero, then x is equal to the mean. Very few values in the distribution have a Z-score less than -3 or greater than +3, especially if the distribution is bell-shaped.
Empirical Rule
allows us to estimate the probability that a randomly selected value in a normal distribution is within a certain interval. This is equivalent to estimating the percentage of values in a normal distribution that are within a specific interval. This enables us to determine whether or not a value in a normal distribution is unusual.
total area under a probability curve
always 1
probability distribution
describes how values vary in a distribution and enables us to find probabilities associated with intervals of values.
A value in a normal distribution is unusual if
it is more than two standard deviations from the mean. (Such a value is outside the middle 95% of the values in the distribution. Only 5% of values in a normal distribution have this property.)
μ
population mean
σ
standard deviation of a population (Sigma) is a measurement of variability,
