AMU MATH 120 Intro to statistics Homework week 2 American Military University

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1. Consider the mode, median, and mean.

(a) Which average represents the middle value of a data distribution?

• mode
• median
• mean

(b) Which average represents the most frequent value of a data distribution?

• mode
• median
• mean

(c) Which average takes all the specific values into account?

• Mode
• median
• mean

2. What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

• statistic, x parameter, x
• statistic, x ; parameter, μ
• statistic, μ; parameter, x
• statistic, μ; parameter, μ

3. Find the mean, median, and mode of the data set.

9 4 8 4 7

• mean (Enter a number.)

• median (Enter an exact number.)

• mode (Enter an exact number.)

4. How large is a wolf pack? The following information is from a random sample of winter wolf packs. Winter pack size are given below. Compute the mean, median, and mode for the size of winter wolf packs. (For each answer, enter a number. Round your answers to one decimal place.)

3 11

8 6 8 8 3 5 4

14

4 16 5 5 3 9 8 9

• Mean
• median
• mode

5. Find the weighted average of a data set where 30 has a weight of 3, 40 has a weight of 5, and 50 has a weight of 2. (Enter an exact number.)

6. What symbol is used for the standard deviation when it is a sample statistic? What symbol is used for the standard deviation when it is a population parameter?

• Sample statistic: σ. Population parameter: s.

• For both, s is used.

• For both, σ is used.

• Sample statistic: s. Population parameter: σ

7. Consider the data set.

2, 3, 4, 5, 8

(a) Find the range. (Enter an exact number.)

(b) Use the defining formula to compute the sample standard deviation s. (Enter a number. Round your answer to two decimal places.)

(c) Use the defining formula to compute the population standard deviation σ. (Enter a number. Round your answer to two decimal places.)

8.

One indicator of an outlier is that an observation is more than 2.5 standard deviations from the mean. Consider the data value 80.

(a) If a data set has mean 70 and standard deviation 5, is 80 a suspect outlier?

• No, since 80 is more than 2.5 standard deviations above the mean.
• Yes, since 80 is more than 2.5 standard deviations above the mean.
• No, since 80 is less than 2.5 standard deviations above the mean.
• Yes, since 80 is less than 2.5 standard deviations above the mean.

(b) If a data set has mean 70 and standard deviation 3, is 80 a suspect outlier?

• No, since 80 is more than 2.5 standard deviations above the mean.
• Yes, since 80 is more than 2.5 standard deviations above the mean.
• No, since 80 is less than 2.5 standard deviations above the mean.
• Yes, since 80 is less than 2.5 standard deviations above the mean.

9.  Given the sample data.

x: 23, 19, 13, 32, 27 (a) Find the range. (Enter an exact number.)

(b) Verify that Σx = 114 and Σx

Σx =

Σx2

For each answer, enter an exact number.)

(c) Use the results of part (b) and appropriate computation formulas to compute the sample variance s 2 and sample standard deviation s. (For each answer, enter a number. Round your answers to two decimal places.)

s 2 =

s=

(d) Use the defining formulas to compute the sample variance s 2 and sample standard deviation s. (For each answer, enter a number. Round your answers to two decimal places.)

s 2 =

 s=

(e) Suppose the given data comprise the entire population of all x values. Compute the population variance σ 2 and population standard deviation σ. (For each answer, enter a number. Round your answers to two decimal places.)

σ 2

 σ=

10. Consider sample data with x = 6 and s = 3. (For each answer, enter an exact number.)

(a) Compute the coefficient of variation (as a percent). %

(b) Compute a 75% Chebyshev interval around the sample mean.

Lower Limit

Upper Limit

11. Describe the relationship between two variables when the correlation coefficient r is one of the following.

(a) near −1

• strong positive linear correlation
• weak negative linear correlation
• weak positive linear correlation
• weak or no linear correlation
• strong negative linear correlation

(b) near 0

• weak positive linear correlation
• strong negative linear correlation
• weak negative linear correlation
• weak or no linear correlation
• strong positive linear correlation

(c) near 1

• strong negative linear correlation
• weak negative linear correlation
• strong positive linear correlation
• weak or no linear correlation
• weak positive linear correlation

12.  Look at the following diagrams. Which diagrams show high linear correlation, moderate or low linear correlation, or no linear correlation? (There are three diagrams. You will be presented the diagrams one at a time.)

• moderate or low
• high
• no

• high
• moderate or low
• no

• no
• high
• moderate or low

13. How much should a healthy Shetland pony weigh? Let x be the age of the pony (in months), and let y be the average weight of the pony (in kilograms).

x	3	6	12	19	25
y	60	95	140	166	185

(a) Make a scatter diagram of the data and visualize the line you think best fits the data. (Submit a file with a maximum size of 1 MB.)

(b) Would you say the correlation is low, moderate, or strong?

• low
• moderate
• strong

Would you say the correlation is positive or negative?

• positive
• negative

(c) Use a calculator to verify that Σ(x) = 65, Σ(x 2 ) = 1175, Σ(y) = 646, Σ(y 2 ) = 94,006, and Σ(x y) = 10,209.

Compute r. (Enter a number. Round your answer to three decimal places.)

As x increases from 3 to 25 months, does the value of r imply that y should tend to increase or decrease? Explain your answer.

• Given our value rofy , should tend to increase xas increases.
• Given our value rofy , should tend to remain constant xas increases.
• Given our value rof , we can not draw any conclusions for the behavior yof xas increases.
• Given our value rofy , should tend to decrease xas increases.

14. Do larger universities tend to have more property crime? University crime statistics are affected by a variety of factors. The surrounding community, accessibility given to outside visitors, and many other factors influence crime rate. Let x be a variable that represents student enrolment (in thousands) on a university campus, and let y be a variable that represents the number of burglaries in a year on the university campus. A random sample of n = 8 universities in California gave the following information about enrolments and annual burglary incidents.

X	13.9	28.2	24.5	14.3	7.5	27.7	16.2	20.1
Y	24	73	39	23	15	30	15	25

(a) Make a scatter diagram of the data. Then visualize the line you think best fits the data. (Submit a file with a maximum size of 1 MB.)

(b) Use a calculator to verify that Σ(x) = 152.4, Σ(x 2 ) = 3283.18, Σ(y) = 244, Σ(y 2 ) = 9930 and Σ(x y) = 5365.6.

Compute r. (Enter a number. Round to 3 decimal places.)

As x increases, does the value of r imply that y should tend to increase or decrease? Explain your answer.

• Given our value rofy , should tend to decrease xas increases.

• Given our value rofy , should tend to increase xas increases.

• Given our value rofy , should tend to remain constant xas increases.

• Given our value rof , we cannot draw any conclusions for the behavior yof xas increases.

15. In the least­squares line ŷ = 5 − 9x, what is the value of the slope? (Enter an exact number.)

When x changes by 1 unit, by how much does ŷ change?

• When x increases by 1 unit, ŷ decreases by −9 units.
• When x increases by 1 unit, ŷ increases by 9 units.
• When x increases by 1 unit, ŷ decreases by 9 units.
• When x decreases by 1 unit, ŷ decreases by 9 units.

16.  If two variables have a negative linear correlation, is the slope of the least­squares line positive or negative?

• negative
• positive

17. You are the foreman of the Bar­S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some calves to add to the Bar­S herd. How much should a healthy calf weigh? Let x be the age of the calf (in weeks), and let y be the weight of the calf (in kilograms).

X	1	5	11	16	26	36
y	39	47	73	100	150	200

Complete parts (a) through (e), given

Σx = 95, Σy = 609, Σx2= 2375, Σy 2 = 81,559, Σx y = 13,777, and r ≈ 0.997.

(b) Verify the given sums Σx, Σy, Σx 2 , Σy 2 , Σx y, and the value of the sample correlation coefficient r. (For each answer, enter a number. Round your value for r to three decimal places.)

Σx =

Σy =

Σy 2

Σx

Σx y =

r=

(c) Find x , and y . Then find the equation of the least­squares line = a + b x. (For each answer, enter a number. Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

(d) Graph the least­squares line. Be sure to plot the point ( x , y ) as a point on the line. (Select the correct graph.)

18. Consider the following ordered data.

2 5 5 6 7 7 8 9 10 (a) Find the low, Q 1 , median, Q 3 , and high. (For each answer, enter a number.)

• low
• Q 1
• median
• Q 3
• high

(b) Find the interquartile range. (Enter a number.)

(c) Make a box­and­whisker plot. (Select the correct graph.)

19. At one hospital there is some concern about the high turnover of nurses. A survey was done to determine how long (in months) nurses had been in their current positions. The responses (in months) of 20 nurses were as follows.

31 10 13 22 33 44 35 50 20 16

15 31 37 34 36 19 28 39 16 44

Make a box­and­whisker plot of the data. (Select the correct graph.)

Find the interquartile range.

(Enter an exact number.) IQR =