Measures of the Location of the Data

Transcription

1 Measures of the Location of the Data Mark has 51 films in his collection. Each movie comes with a rating on a scale from 0.0 to The following table displays the ratings of the aforementioned films in the increasing order: Find the median M, the first quartile Q1, the third quartile Q3, the interquartile range IQR. 2. Compute the two numbers: Q IQR and Q IQR Are there any potential outliers in the given data? 3. Find the following percentiles: (a) The 19 th percentile. (b) The 42 nd percentile. (c) The 65 th percentile. (d) The 90 th percentile. 4. To what percentiles belong the movies with the following ratings? (a) 4.9 (b) 5.9 (c) 7.0 (d) Draw a box-whisker plot for the given data. (Start by drawing a scaled number line!) 6. Twelve students were asked how many movies they saw last month. Here are their responses: 4, 9, 15, 3, 7, 1, 9, 12, 10, 8, 5, 7. Find the following: (a) The mean (b) The median (c) The mode(s)

2 7. The following frequency table shows the distribution of scores obtained by shooters in a sports competition: Class Interval Frequency (a) Find the mean of this grouped frequency table. (b) Estimate the median. (c) What is the midrange of the data? (d) Based on your computations, how is the distribution of data skewed? 8. Two hundred and seventeen households were surveyed and the number of plants in each household was recorded in the form of the following frequency table: Number of plants Frequency (a) What is the mean? (b) What is the median? (c) What is (are) the mode(s)? (d) Which of the above measures of the center is the largest? Which is the smallest? (e) How are the data skewed? 9. The following data reflects the number of cars that had parked on certain a parking lot during six consecutive days, Monday through Saturday: { 43, 52, 42, 48, 34, 21 }. (a) Find the average number of cars. (b) Find the variance. (c) Find the standard deviation.

3 10. Five runners showed the following results on a distance of 400 meters, in seconds: (a) What is the mean of their results? (b) What is the value of the variance? (c) What is the value of the standard deviation? (d) What is the z-score of the runner with a result of 67.0 seconds? (e) What is the z-score of the runner with a result of 54.7 seconds? 11. Tim Hortons has asked a random sample of 15 people how many coffees they buy per month. Here are the results: Compute the following: (a) the mean (b) the variance (c) the standard deviation 12. A movie theater is interested in the average length of a film. They randomly sampled several films and obtained a mean of 100 minutes and a standard deviation of 7 minutes. (a) Use Chebyshev s Theorem to construct an interval that will contain (approximately) at least 75% (b) Use Chebyshev s Theorem to construct an interval that will contain (approximately) at least 89% (c) Use Chebyshev s Theorem to construct an interval that will contain (approximately) at least 95% 13. Consider the following frequency table for the age of workers in a company: Class Interval, in years Frequency (a) Estimate the mean age of workers in this company. (b) Compute the standard deviation for this frequency table.

4 14. The following table gives a frequency distribution for the weekly grocery bill in 100 randomly selected households of two adults and two children. Class Interval $50 - $100 $100 - $150 $150 - $200 $200 - $250 $250 - $300 Frequency Find (a) The mean weekly grocery bill for these households. (b) The standard deviation for these households. 15. One group of 50 runners on a track distance of 10,000 meters showed an average time of 36.2 minutes with a standard deviation of 6.8 minutes. Another group of 50 runners on the same distance showed an average time of 38.6 minutes with a standard deviation of 3.5 minutes. Adam ran in the first group and showed a time of 32.0 minutes. Brett ran in the second group and finished with a time of 34.4 minutes. If each group was ranked from 1 to 50 from the fastest to the slowest runner, which of the two runners, Adam or Brett, would have a better ranking?

5 ANSWERS. M 7.2, Q 6.4, Q 8.2, IQR Q1-1.5 IQR = 3.7 and Q IQR = There are two potential lower outliers: 2.7 and (a) 6.0 (b) 7.0 (c) 7.55 (d) (a) 7 th (b) 14 th (c) 43 rd (d) 85 th (a) 7.5 (b) 7.5 (c) 7 and 9 7. (a) x = (b) M 24.4 (c) 18.4 (d) Since the mean is smaller than the median, and both are to the right of the midrange of the data, the distribution of data is skewed to the left. This can also be confirmed by visualizing the data using, for example, a histogram. 8. (a) x = 3.9 (b) M = 3 (c) Mode = 2 (d) The mean is the largest, while the mode is the smallest. (e) Since the mean is the largest, and the mode is the smallest, the distribution of data appears to be skewed to the right. This can also be confirmed by visualizing the data using, for example, a histogram. 9. (a) x = 40 cars (b) s 2 = (c) s = 11.1 cars 10. (a) x = seconds (b) s 2 = (c) s = 6.26 seconds (d) z = 1.28 (e) z = (a) 19.7 cups (b) (c) 16.5 cups 12. The intervals (in minutes) are: (a) [86, 114] (b) [79, 121] (c) [68.5, 131.5] 13. (a) 45.8 years (b) 10.6 years 14. (a) $173 (b) $ Brett would be ranked higher, because his z-score (z B = 1.2) is smaller than Adam s (z A = 0.62), meaning that his number in ranking is smaller than Adam s.