Section 4 Homework

Find the range of the data set represented by the graph (since I don't have Quizlet+, I can't insert the image of the graph; ergo, I pasted the description). A graph titled *"Woman's Age at First Childbirth"* has a horizontal axis labeled *Age (in years)* from *24 to 34* in *increments of "1"*, and a vertical axis labeled *Frequency* from *0 to 10* in *increments of "2"*. The graph contains vertical bars of width 1, where one vertical bar is centered over each of the horizontal axis tick marks. The heights of the vertical bars are listed as follows, where the *label* is listed *first*, and the *height* is listed *second*: *(24, 2); (25, 2); (26, 5); (27, 6); (28, 7); (29, 10); (30, 9); (31, 6); (32, 4); (33, 2); (34, 2)*. The range of the data set is *__*.

Correct Answer: *10*

Find the range of the data set represented by the graph (*simplify* the answer). (Since I don't have Quizlet+, I can't insert the image of the dot plot; ergo, I pasted the description.) A dot plot has a horizontal axis labeled from *70 to 100* in *increments of "1"*. The graph consists of a series of plotted points from left to right. The coordinates of the plotted points are as follows, where the *label* is listed *first*, and the *number of dots* is listed *second*: *(75, 1); (76, 2); (79, 1); (80, 4); (83, 1); (86, 2); (88, 4); (89, 2); (90, 2); (93, 1); (94, 1); (95, 1); (96, 1); (100, 1)*. The range of the data set is *__*.

Correct Answer: *25*

Use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 67 miles per​ hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.​ (Assume the data set has a​ bell-shaped distribution.) Approximately *__%* of vehicles travel between 63 miles per hour and 71 miles per hour.

Correct Answer: *68%*

Both data sets have a mean of 245. One has a standard deviation of​ 16, and the other has a standard deviation of 24. Key: *20* | *8* = *208* *(a)*: *20* | *8* *9* *21* | *2* *5* *8* *22* | *1* *3* *23* | *0* *0* *6* *7* *24* | *2* *5* *8* *25* | *1* *3* *6* *8* *26* | *0* *7* *9* *27* | *8* *28* | *3* *5* *7* *(b)*: *20* | *21* | *3* *22* | *2* *3* *5* *23* | *0* *4* *5* *6* *8* *24* | *1* *1* *2* *3* *3* *3* *25* | *1* *5* *8* *8* *26* | *2* *3* *4* *5* *27* | *0* *7* *28* | Which data set has which​ deviation? A.) *(a)* has a standard deviation of 24 and​ *(b)* has a standard deviation of​ 16, because the data in​ *(a)* have more variability. B.) *​(a)* has a standard deviation of 16 and​ *(b)* has a standard deviation of ​ 24, because the data in​ *(b)* have less variability.

Correct Answer: A.) *(a)* has a standard deviation of 24 and​ *(b)* has a standard deviation of​ 16, because the data in​ *(a)* have more variability.

You are applying for a job at two companies. Company A offers starting salaries with *μ* = $26,000 and *σ* =$4,000. Company B offers starting salaries with *μ* = $26,000 and *σ* = $9,000. From which company are you more likely to get an offer of ​$34,000 or​ more? Choose the correct answer below. A.) Company​ B, because data values that lie within one standard deviation from the mean are not considered unusual. B.) Company​ A, because data values that lie more than two standard deviations from the mean are considered unusual. C.) No​ difference, because data values that lie more than three standard deviations from the mean are considered very unusual.

Correct Answer: A.) Company​ B, because data values that lie within one standard deviation from the mean are not considered unusual.

Why is the standard deviation used more frequently than the​ variance? Choose the correct answer below. A.) The standard deviation is easier to compute. B.) The units of variance are squared. Its units are meaningless. C.) The standard deviation requires less entries from the data set.

Correct Answer: B.) The units of variance are squared. Its units are meaningless.

The English statistician Karl Pearson​ (1857-1936) introduced a formula for the skewness of a distribution. *P* = (3×(x̄ − median))/s Most distributions have an index of skewness between −3 and 3. When P​ > 0 the data are skewed right. When P​ < 0 the data are skewed left. When P​ = 0 the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each. *Part 1 (a):* The coefficient of skewness for x̄ ​= 17​, s​ = 2.7​, median​ = 19, is P​ = *___*. ​(Round to the *nearest hundredth* (*two* decimal places).) *Part 2 (a):* Describe the shape of the distribution. A.) The data are skewed right. B.) The data are symmetric. C.) The data are skewed left. *Part 3 (b):* The coefficient of skewness for x̄ ​= 34​, s​ = 5.4​, median​ = 29 is P​ = *___*. ​(Round to the *nearest hundredth* (*two* decimal places).) *Part 4 (b):* Describe the shape of the distribution. A.) The data are skewed left. B.) The data are skewed right. C.) The data are symmetric.

Correct Answers: *Part 1 (a):* *-2.22* *Part 2 (a):* C.) The data are skewed left. *Part 3 (b):* *2.78* *Part 4 (b):* B.) The data are skewed right.

You used *SS*∨x = ∑×(x − x̄)² when calculating variance and standard deviation. An alternative formula for the standard deviation that is sometimes convenient for hand calculations is shown below. You can find the sample variance by dividing the sum of squares by n−​1, and the sample standard deviation by finding the square root of the sample variance. Complete parts​ (a) and​ (b) below. *SS*∨x = ∑x² − (((∑x)²)/n) *Shoppers' Ages*: *20 14 20 20 19 17 15 19 20 16 19 16 20 16 15 18 14 23 19 19* *Part 1 (a):* Use the shortcut formula to calculate the sample standard deviation. (*Simplify* the answer, and round to *one* decimal place.) s = *__* *Part 2 (b):* Find the sample standard deviation using the usual formula of s = √(∑(x − x̄)²/n−1​), and compare it to your result from *part 1 (a)*. (*Simplify* the answer, and round to *one* decimal place.) The sample standard deviation is *_(1)_*. This is *____(2)____* the result from *part 1 (a)*.

Correct Answers: *Part 1 (a):* *2.4* *Part 2:* *(1):* *2.4* *(2):* *equal to*

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed. *41 44 43 62 28 28 41 44 43 26 62 41 51* ​ *(a):* Find the sample mean and sample standard deviation. (Parts 1 & 2) *​(b):* Each employee in the sample is given a 6​% raise. Find the sample mean and sample standard deviation for the revised data set. (Parts 3 & 4) ​*(c):* To calculate the monthly​ salary, divide each original salary by 12. Find the sample mean and sample standard deviation for the revised data set. (Parts 5 & 6) ​*(d):* What can you conclude from the results of​ *parts 1 & 2 (a)*, *parts 3 & 4 (b)*, and​ *parts 5 & 6 (c)*? (Part 7) *Part 1 (a):* The sample mean is x̄ = *___* thousand dollars. ​(Round to *one* decimal place.) *Part 2 (a):* The sample standard deviation is s = *___* thousand dollars. ​(Round to *one* decimal place.) *Part 3 ​(b):* The sample mean is x̄ = *___* thousand dollars. ​(Round to *one* decimal place.) *Part 4 (b):* The sample standard deviation is s = *___* thousand dollars. ​(Round to *one* decimal place.) *Part 5 ​(c):* The sample mean is x̄ = *__* thousand dollars. ​(Round to *one* decimal place.) *Part 6 (c):* The sample standard deviation is s = *__* thousand dollars. ​(Round to *one* decimal place.) *Part 7 ​(d):* What can you conclude from the results of​ *parts 1 & 2 (a)*, *parts 3 & 4 (b)*, and​ *parts 5 & 6 (c)*? A.) When each entry is multiplied by a constant​ *k*, the new sample mean is *k × x̄* and the sample standard deviation remains unaffected. B.) When each entry is multiplied by a constant​ *k*, the sample mean and the sample standard deviation remain unaffected. C.) When each entry is multiplied by a constant​ *k*, the new sample standard deviation is *k × s* and the sample mean remains unaffected. D.) When each entry is multiplied by a constant​ *k*, the new sample mean is *k × x̄* and the new sample standard deviation is *k × s*.

Correct Answers: *Part 1 (a):* *42.6* *Part 2 (a):* *11.3* *Part 3 (b):* *45.2* *Part 4 (b):* *12.0* *Part 5 (c):* *3.6* *Part 6 (c):* *0.9* *Part 7 (d):* D.) When each entry is multiplied by a constant​ *k*, the new sample mean is *k × x̄* and the new sample standard deviation is *k × s*.

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed. *55 45 49 61 30 30 55 45 49 29 61 55 52* *(a):* Find the sample mean and sample standard deviation. (Parts 1 & 2) *(b):* Each employee in the sample is given a ​$5000 raise. Find the sample mean and sample standard deviation for the revised data set. (Parts 3 & 4) *(c):* Each employee in the sample takes a pay cut of ​$2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (Parts 5 & 6) ​*(d):* What can you conclude from the results of​ *parts 1 & 2 (a)*, *parts 3 & 4 (b)*, and​ *parts 5 & 6 (c)*? (Part 7) *Part 1 ​(a):* The sample mean is x̄ = *___* thousand dollars. (Round to *one* decimal place.) *Part 2 (a):* The sample standard deviation is s = *___* thousand dollars. (Round to *one* decimal place.) *Part 3 ​(b):* The sample mean is x̄ = *___* thousand dollars. (Round to *one* decimal place.) *Part 4 (b):* The sample standard deviation is s = *___* thousand dollars. (Round to *one* decimal place.) *Part 5 ​(c):* The sample mean x̄ = *___* thousand dollars. (Round to *one* decimal place.) *Part 6 (c):* The sample standard deviation is s = *___* thousand dollars. (Round to *one* decimal place.) *Part 7 ​(d):* What can you conclude from the results of​ *parts 1 & 2 (a)*, *parts 3 & 4 (b)*, and​ *parts 5 & 6 (c)*? A.) When a constant *k* is added to or subtracted from each​ entry, the new sample mean is *x̄ + k* or *x̄ − ​k*, ​respectively, and the new sample standard deviation is *s + k*. B.) When a constant *k* is added to or subtracted from each​ entry, the new sample mean is *x̄ + k* or *x̄ − ​k*, ​respectively, and the sample standard deviation remains unaffected. C.) When a constant *k* is added to or subtracted from each​ entry, the sample mean is​ unaffected, and the new sample standard deviation is *s + k* or *s−​k*, respectively. D.) When a constant *k* is added to or subtracted from each​ entry, the new sample mean is *x̄ + k* or *x̄ − ​k*, ​respectively, and the new sample standard deviation is *s × k*

Correct Answers: *Part 1 (a):* *47.4* *Part 2 (a):* *11.3* *Part 3 (b):* *52.4* *Part 4 (b):* *11.3* *Part 5 (c):* *45.4* *Part 6 (c):* *11.3* *Part 7 (d):* B.) When a constant *k* is added to or subtracted from each​ entry, the new sample mean is *x̄ + k* or *x̄ − ​k*, ​respectively, and the sample standard deviation remains unaffected.

The mean value of land and buildings per acre from a sample of farms is ​$1700​, with a standard deviation of ​$300. The data set has a​ bell-shaped distribution. Assume the number of farms in the sample is 74. *Part 1 (a):* Use the empirical rule to estimate the number of farms whose land and building values per acre are between ​$1400 and ​$2000 (round to the *nearest "whole" number*). *__* farms *Part 2 (b):* If 29 additional farms were​ sampled, about how many of these additional farms would you expect to have land and building values between ​$1400 per acre and ​$2000 per​ acre (round to the *nearest "whole" number*)? *__* farms out of 29

Correct Answers: *Part 1 (a):* *51* *Part 2 (b):* *20*

The distribution of the number of hours that a random sample of people spend doing chores per week is shown in the pie chart. Use 32 as the midpoint for​ "30+ hours." Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. A circle graph titled *"Weekly Chore Hours"* is divided into seven sectors with labels and approximate sizes as numbers of people as follows: *0-4 hours (4 people); 5-9 hours (13 people); 10-14 hours (26 people); 15-19 hours (18 people); 20-24 hours (14 people); 25-29 hours (10 people); 30+ hours (6 people)*. *Part 1:* First construct the frequency distribution. Class = *​Frequency, f* ​0-4 = *_(1)_* ​5-9 = *_(2)_* ​10-14 = *_(3)_* ​15-19 = *_(4)_* ​20-24 = *_(5)_* ​25-29 = *_(6)_* ​30+ = *_(7)_* *Part 2:* Find an approximation for the sample mean. (Type answer as either an *integer* or a *decimal* rounded to the *nearest tenth* (*one* decimal place).) x̄ = *___* *Part 3:* Find an approximation for the sample standard deviation. (Type answer as either an *integer* or a *decimal* rounded to the *nearest tenth* (*one* decimal place).) s = *__*

Correct Answers: *Part 1:* *(1):* *4* *(2):* *13* *(3):* *26* *(4):* *18* *(5):* *14* *(6):* *10* *(7):* *6* *Part 2:* *16.3* *Part 3:* *7.8*

Another measure of variation is the Mean Absolute Deviation. It is computed using the formula *M.A.D.* = (*∑*×|*x*∨i − *x̄*|)/n. Compute the mean absolute deviation of the sample data shown below and compare the results with the sample standard deviation. ​ *$980*, *$2,035*, *​$918*, *$1,893* *Part 1:* The *mean absolute deviation* of the data is *$_____*. ​(Round to *two* decimal places.) *Part 2:* The *sample standard deviation* of the data is *$_____*. ​(Round to *two* decimal places.) *Part 3:* The mean absolute deviation of the data is *__________* the sample standard deviation of the data.

Correct Answers: *Part 1:* *$507.50* *Part 2:* *$589.41* *Part 3:* *less than*

The estimated distribution​ (in millions) of the population by age in a certain country for the year 2015 is shown in the pie chart. Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set (round to *two* decimal places for both). Use 70 as the midpoint for​ "65 years and​ over." An untitled circle graph is divided into eight sectors with labels and approximate sizes as a percentage of a circle as follows: *Under 4 years, 23.1, (7%); 5-14 years, 43.5, (13%); 15-19 years, 17.3, (5%); 20-24 years, 23.3, (7%); 25-34 years, 52, (15%); 35-44 years, 41.4, (12%); 45-65 years, 81.5, (24%); 65 years and over, 54.6, (16%)*. *Part 1:* The sample mean is *x̄* = *____*. ​ *Part 2:* The sample standard deviation is *s* = *____*.

Correct Answers: *Part 1:* *37.72* *Part 2:* *21.89*

The ages​ (in years) and heights​ (in inches) of all pitchers for a baseball team are listed. Find the coefficient of variation for each of the two data sets. Then compare the results. *Heights* = *Ages* *78* = *22* *79* = *23* *71* = *24* *71* = *26* *79* = *26* *78* = *28* *75* = *30* *80* = *29* *74* = *31* *73* = *32* *71* = *32* *72* = *31* *Part 1:* *CV*∨heights = *__%* ​(Round to *one* decimal place.) *Part 2:* *CV*∨ages = *___%* ​(Round to *one* decimal place.) *Part 3:* Compare the results. What can you​ conclude? A.) Ages are more variable than heights for all pitchers on this team. B.) Heights are more variable than ages for all pitchers on this team. C.) Ages and heights for all pitchers on this team have about the same amount of variability.

Correct Answers: *Part 1:* *4.5%* *Part 2:* *12.3%* *Part 3:* A.) Ages are more variable than heights for all pitchers on this team.

The coefficient of variation CV describes the standard deviation as a percent of the mean. Because it has no​ units, you can use the coefficient of variation to compare data with different units. Find the coefficient of variation for each sample data set. What can you​ conclude? *CV* = ((Standard deviation)/(Mean))×100% *Heights* = *Weights* *76* = *197* *66* = *210* *67* = *223* *72* = *184* *73* = *221* *71* = *183* *66* = *183* *77* = *167* *66* = *223* *80* = *167* *79* = *230* *73* = *173* *Part 1:* *CV*∨heights = *__%* ​(Round answer to the *nearest tenth* (*one* decimal place).) *Part 2:* *CV*∨weights = *___%* ​(Round answer to the *nearest tenth* (*one* decimal place).) *Part 3:* What can you​ conclude? A.) Height is more variable than weight. B.) Weight is more variable than height.

Correct Answers: *Part 1:* *7.1%* *Part 2:* *12.0%* *Part 3:* B.) Weight is more variable than height.

Explain how to find the range of a data set. What is an advantage of using the range as a measure of​ variation? What is a​ disadvantage? *Part 1:* Explain how to find the range of a data set. Choose the correct answer below. A.) The range is found by subtracting the minimum data entry from the maximum data entry. B.) The range is found by adding the first and last data entries. C.) The range is found by subtracting the first data entry from the last data entry. D.) The range is found by adding the minimum and maximum data entries. *Part 2:* What is an advantage of using the range as a measure of​ variation? A.) It uses all entries from the data set. B.) It uses only two entries from the data set. C.) It is easy to compute. *Part 3:* What is a disadvantage of using the range as a measure of​ variation? A.) It uses only two entries from the data set. B.) It is hard to compute. C.) It uses all entries from the data set.

Correct Answers: *Part 1:* A.) The range is found by subtracting the minimum data entry from the maximum data entry. *Part 2:* C.) It is easy to compute. *Part 3:* A.) It uses only two entries from the data set.