Homework 3.4/3.5 Measures of Position/Quartiles & Outliers

Find the​ five-number summary, and​ (b) draw a​ box-and-whisker plot of the data. 3 8 8 6 1 9 8 7 9 6 9 4 1 6 2 9 8 7 7 9

1, 5, 7, 8.5, 9 should look like a left skewed box-whisker plot

A __________ represent the number of standard deviations a data value is from the mean.

Z-score

Find the​ five-number summary, and​ (b) draw a​ box-and-whisker plot of the data. 4 8 8 5 2 9 8 7 9 5 9 4 2 5 2 9 8 7 7 9

2, 4.5, 7, 8.5, 9 should look like a left skewed box-whisker plot

_______ divide data sets in fourths.

Quartiles (The most common percentiles are quartiles. Quartiles divide data sets into​ fourths, or four equal parts. The first​ quartile, denoted Q1​, divides the bottom​ 25% of the data from the top​ 75%. Therefore, the first quartile is equivalent to the 25th percentile. The second​ quartile, Q2​, divides the bottom​ 50% of the data from the top​ 50%; it is equivalent to the 50th percentile or the median.​ Finally, the third​ quartile, Q3​, divides the bottom​ 75% of the data from the top​ 25%; it is equivalent to the 75th percentile.)

The​ _______ represents the number of standard deviations an observation is from the mean.

Z-score (he​ z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. It is found by subtracting the mean from the data value and dividing the result by the standard deviation. The​ z-score is unitless. It has mean 0 and standard deviation 1.)

Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 3000 grams and a standard deviation of 900 grams while babies born after a gestation period of 40 weeks have a mean weight of 3600 grams and a standard deviation of 590 grams. If a 35​-week gestation period baby weighs 2775 grams and a 40​-week gestation period baby weighs 3375 ​grams, find the corresponding​ z-scores. Which baby weighs less relative to the gestation​ period?

35-week: -0.25 40-Week: -0.38 40-week baby weighs less

A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 4 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 7 cm with a standard deviation of 0.10 cm. For what lengths will a bolt be​ destroyed?

<6.6 and >7.4 will be destroyed

Which of the accompanying boxplots likely has the data with the larger standard​ deviation? Why?

Boxplot II likely has the data with the larger standard deviation because the boxplot appears to have a greater​ spread, which likely results in a larger standard deviation

In a​ boxplot, if the median is to the left of the center of the box and the right whisker is substantially longer than the left​ whisker, the distribution is skewed ___________

Right

The mean finish time for a yearly amateur auto race was 186.51 minutes with a standard deviation of 0.378 minute. The winning​ car, driven by Roger​, finished in 185.53 minutes. The previous​ year's race had a mean finishing time of 112.8 with a standard deviation of 0.125 minute. The winning car that​ year, driven by Tammy​, finished in 112.56 minutes. Find their respective​ z-scores. Who had the more convincing​ victory?

Roger: -2.59 Tammy: -1.92 Roger had a more convincing victory because of a lower​ z-score

The data represent the age of world leaders on their day of inauguration. Find the​ five-number summary, and construct a boxplot for the data. Comment on the shape of the distribution. 67 57 50 48 67 54 53 51 43 53 50 61 47 46 46

The​ five-number summary is 43​, 47​, 51​, 57​, 67. The box-whisker plot right skewed.

Violent crimes include​ rape, robbery,​ assault, and homicide. The following is a summary of the​ violent-crime rate​ (violent crimes per​ 100,000 population) for all states of a country in a certain year. Complete parts​ (a) through​ (d). Upper Q1=272.8​, Upper Q2=388.5​, Upper Q3=529.7 Provide an interpretation of these results. Choose the correct answer below. A. ​25% of the states have a​ violent-crime rate that is 272.8 crimes per​ 100,000 population or more.​ 50% of the states have a​ violent-crime rate that is 388.5 crimes per​ 100,000 population or more.​ 75% of the states have a​ violent-crime rate that is 529.7 crimes per​ 100,000 population or more. B. ​25% of the states have a​ violent-crime rate that is 272.8 crimes per​ 100,000 population or less.​ 50% of the states have a​ violent-crime rate that is 388.5 crimes per​ 100,000 population or less.​ 75% of the states have a​ violent-crime rate that is 529.7 crimes per​ 100,000 population or less. C. ​75% of the states have a​ violent-crime rate that is 272.8 crimes per​ 100,000 population or less.​ 50% of the states have a​ violent-crime rate that is 388.5 crimes per​ 100,000 population or less.​ 25% of the states have a​ violent-crime rate that is 529.7 crimes per​ 100,000 population or less.

a) 25% of the states have a​ violent-crime rate that is 272.8 crimes per​ 100,000 population or less.​ 50% of the states have a​ violent-crime rate that is 388.5 crimes per​ 100,000 population or less.​ 75% of the states have a​ violent-crime rate that is 529.7 crimes per​ 100,000 population or less. b) The interquartile range is 256.9 crimes per​ 100,000 population. c) The middle​ 50% of all observations have a range of 256.9 crimes per​ 100,000 population. d) The lower fence is -112.55 crimes per​ 100,000 population. The upper fence is 915.05 crimes per​ 100,000 population e) Yes because it more than the upper fence f) The distribution of​ violent-crime rates is skewed right.

A highly selective boarding school will only admit students who place at least 2 standard deviations above the mean on a standardized test that has a mean of 300 and a standard deviation of 26. What is the minimum score that an applicant must make on the test to be​ accepted?

The minimum score that an applicant must make on the test to be accepted is 352

Explain the meaning of the following percentiles in parts​ (a) and​ (b). ​(a) The 10th percentile of the weight of males 36 months of age in a certain city is 13.0 kg. ​(b) The 95th percentile of the length of newborn females in a certain city is 54.3 cm.

​a) 10% of​ 36-month-old males weigh 13.0 kg or​ less, and 90​% of​ 36-month-old males weigh more than 13.0 kg. b)​ 95% of newborn females have a length of 54.3 cm or​ less, and 5​% of newborn females have a length that is more than 54.3 cm

One year Dan had the lowest ERA​ (earned-run average, mean number of runs yielded per nine innings​ pitched) of any male pitcher at his​ school, with an ERA of 3.35. ​Also, Rita had the lowest ERA of any female pitcher at the school with an ERA of 3.42. For the​ males, the mean ERA was 3.902 and the standard deviation was 0.936. For the​ females, the mean ERA was 4.471 and the standard deviation was 0.748. Find their respective​ z-scores. Which player had the better year relative to their​ peers, Dan or Rita​? ​(Note: In​ general, the lower the​ ERA, the better the​ pitcher.)

Dan had an ERA with a​ z-score of -0.59. Rita had an ERA with a​ z-score of -1.41. Rita had a better year because of a lower​ z-score.

Explain the circumstances for which the interquartile range is the preferred measure of dispersion. What is an advantage that the standard deviation has over the interquartile​ range? A. The interquartile range is preferred when the data are skewed or have outliers. An advantage of the standard deviation is that it uses all the observations in its computation. B. The interquartile range is preferred when the data are not skewed or no have outliers. An advantage of the standard deviation is that it uses all the observations in its computation. C. The interquartile range is preferred when the distribution is symmetric. An advantage of the standard deviation is that it increases as the dispersion of the data increases. D. The interquartile range is preferred when the data are bell shaped. An advantage of the standard deviation is that it increases as the dispersion of the data increases. E. The interquartile range is preferred when the data are bell shaped. An advantage of the standard deviation is that it is resistant to extreme values. F. The interquartile range is preferred when the distribution is symmetric. An advantage of the standard deviation is that it is resistant to extreme values.

The interquartile range is preferred when the data are skewed or have outliers. An advantage of the standard deviation is that it uses all the observations in its computation. (The interquartile​ range, IQR, is the range of the middle​ 50% of the observations in a data set. That​ is, the IQR is the difference between the first and third quartiles. The interquartile range is not affected by extreme values. The standard deviation describes how​ far, on​ average, each observation is from the mean. It is affected by extreme values.)