Percentiles
Z Scores
Normal Distributions
MISC
100
The cumulative relative frequency graph shows the distribution of median household incomes for the 50 states and the District of Columbia in a recent year. Use the cumulative relative frequency graph for the state income data to answer each question. At what percentile is California, with a median household income of $57,445?
California is at about the 78th percentile for household income
100
In 2016, the mean number of wins for teams in Major League Baseball was 81 wins with a standard deviation of 10.7 wins.
The Chicago Cubs broke the Curse of the Billy Goat by winning the World Series in 2016. Find and interpret the z-score for the Chicago Cubs, who had 103 wins in 2016.
The number of wins by the Chicago Cubs in 2016 is 2.06 standard deviations above the Major League Baseball mean of 81 wins.
100
(BONUS x2)
A certain density curve is defined by the line segment that connects the points (0, 0) and (4, 0.5).
(a) Draw a picture of the density curve.
(b) What percent of values will fall between 0 and 2?
(a)
(b)
100
What is the largest planet in our solar system?
Jupiter
200
The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2016.
(a) Find the percentile for the Detroit Tigers, who had 86 wins.
(b) The number of wins for the Cleveland Indians is at the 90th percentile of the distribution. Interpret this value in context. How many wins did the Cleveland Indians have in 2016?
(a) 20/30 = 0.67 or the 67th percentile
(b) The Cleveland Indians had 94 wins in the 2016 season
200
In 2016, the mean number of wins for teams in Major League Baseball was 81 wins with a standard deviation of 10.7 wins.
The Chicago White Sox had a z-score of -0.28. Find the number of wins for the Chicago White Sox for 2016.
The Chicago White Sox had 78 wins in the 2016 season.
200
Chapter 1 quiz scores from Mr. Pastori’s 1-2B class follow an approximately Normal distribution with a mean of 81 and standard deviation of 6.
(a) About what percent of students scored less than 93 on the Chapter 1 test?
(b) A student who scored a 68 would be at about what percentile of the distribution?
(a) 97.5% of students scored less than 93
(b) About the 1.5th percentile because about 100% − 98.5% = 1.5% of students scored less than or equal to 68
200
BONUS TOPIC (x2 POINTS)
What Color is Math?
Blue
300
The cumulative relative frequency graph shows the distribution of median household incomes for the 50 states and the District of Columbia in a recent year. Use the cumulative relative frequency graph for the state income data to answer each question. Estimate and interpret the first quartile of this distribution.
The first quartile of this distribution is the 25th percentile. About 25% of states have median incomes less than or equal to $45,000
300
BONUS TOPIC (x2 POINTS)
All 30 guys on a high school track team ran a 1-mile race, and their times were recorded. It was later discovered that the person in charge of the stopwatch did not start the time until 10 seconds into the race. Also, the athletes wondered what these times would look like if they were given in minutes instead of seconds.
What would the Shape, Median, and Standard Deviation be after the transformations?
Skewed Right
Median = (432.85 + 10)/60 = 7.38 minutes
SD = 37.876/60 = 0.631 minutes
300
Find the proportion of observations in a standard Normal distribution that satisfies each of the following statements.
(a) z < −2.46
(b) 0.89 < z < 2.46
(a) Table A: 0.0069;
normalcdf(lower: -1000, upper: -2.46, mean: 0, SD: 1) = 0.0069
(b) Table A: 0.9931 - 0.8133 = 0.1798;
normalcdf(lower: 0.89, upper: 2.46, mean: 0, SD: 1) = 0.1798
300
Which element has the chemical symbol ‘Hg’?
Mercury
400
BONUS TOPIC (x2 POINTS)
Taking an Uber ride in New York City has an initial fee of $2.55 with an additional charge of $1.75 per mile. In equation form, cost = 2.55 + 1.75(miles). A local New York City resident records the number of miles for his first 25 rides with Uber. The mean distance of his rides is 5.6 miles with a standard deviation of 1.2 miles.
(a) Find the mean cost of the 25 trips.
(b) Calculate the standard deviation of the cost of the 25 trips. Interpret this value in context.
(a) Mean = (1.75)(5.6) + 2.55 = $12.35
(b) SD = (1.75)(1.2) = $2.10; the costs of an Uber ride for this resident typically vary from the mean by about $2.10.
400
Three landmarks of baseball achievement are Ty Cobb’s 0.420 batting average, Ted Williams’s 0.406, and George Brett’s 0.309. These batting averages cannot be compared directly because the distribution of major league batting averages has changed over the years. Here are the facts: Decade Mean Standard deviation for each player respectively. Find the standardized scores for Cobb, Williams, and Brett. Who had the best performance for the decade he played?
Cobb = 4.15
Williams = 4.26
Brett = 4.07
All three hitters were at least 4 standard deviations above their peers, but Williams’s z-score is the highest
400
The amount of sleep that elementary school children get per night follows a Normal distribution with a mean of 9.5 hours and a standard deviation of 0.55 hours.
(a) Find the proportion of elementary school children who get between 9 and 10 hours of sleep per night. Sketch the Normal curve and shade the area under the curve that is the answer to the question.
(b) Is it unusual for an elementary school student to get at least 12 hours of sleep? Justify your answer
(a) normalcdf(lower: 9, upper: 10, mean: 9.5, SD: 0.55) = 0.637
About 63.7% of elementary school children get between 9 and 10 hours of sleep per night
(b) normalcdf(lower: 12, upper: 1000, mean: 9.5, SD: 0.55) ≈ 0
It is definitely unusual for an elementary school student to get at least 12 hours of sleep per night. About none of the students get this much sleep per night.
400
What School District did Mr Pastori come from previously?
Elmwood Park
500
Which of the following is closest to the 61st percentile of the standard Normal distribution?
z = 0.28, The percentile in a standard Normal table that is closest to 0.61 is 0.6103, which corresponds to a z-score of 0.28.
500
The amount of time Ricardo spends brushing his teeth follows a Normal distribution with unknown mean and standard deviation. Ricardo spends less than 1 minute brushing his teeth about 40% of the time. He spends more than 2 minutes brushing his teeth 2% of the time. Use this information to determine the mean and standard deviation of this distribution.
mean = 1.1087 minutes
standard deviation = 0.4348 minutes
500
At one particular restaurant, large drink cups require lids with a “diameter” of between 3.95 and 4.05 inches. The restaurant’s lid supplier claims that the diameter of its large lids follows a Normal distribution with mean 3.98 inches and standard deviation 0.02 inch. The supplier is considering two changes to reduce to 1% the percentage of its large-cup lids that are too small. One strategy is to adjust the mean diameter of its lids. Another option is to alter the production process, thereby decreasing the standard deviation of the lid diameters.
(a) If the standard deviation remains at σ = 0.02 inch, at what value should the supplier set the mean diameter of its large-cup lids so that only 1% are too small to fit?
(b) If the mean diameter stays at μ = 3.98 inches, what value of the standard deviation will result in only 1% of lids that are too small to fit?
(c) Which of the two options in parts (a) and (b) do you think is preferable? Justify your answer.
(a) invNorm(area: 0.01, mean: 0, SD: 1) = -2.33. Solving -2.33 = (3.95 - mean)/0.02 gives mean = 4.00 inches.
(b) invNorm(area: 0.01, mean: 0, SD: 1) = -2.33. Solving -2.33 = (3.95 - 3.98)/SD gives SD = 0.013 inch.
(c) We prefer reducing the SD, as in part (b). This will reduce the number of lids that are too small and the number of lids that are too big. If we make the mean a little larger, as in part (a), we will reduce the number of lids that are too small, but we will increase the number of lids that are too big.
500
Nikola Tesla claimed to invent something called Teleforce. What device was this the name of?
Death Beam