Q: Given the following frequency distribution of commute times (in minutes): Time: 0-10 (5), 10-20 (12), 20-30 (8), 30-40 (5) What is the approximate mean commute time using class midpoints?

$\frac{5 \times 5 + 12 \times 15 + 8 \times 25 + 5 \times 35}{5+12+8+5} = 18.5$ minutes

Q: A student receives the following grades: 90 (weight 3), 80 (weight 2), 70 (weight 1). What is the weighted mean of the student's grades?

$\frac{90 \times 3 + 80 \times 2 + 70 \times 1}{3+2+1} = 83.33$

Q: A company produces metal rods with a mean length of 100 cm and a standard deviation of 2 cm. According to the Range Rule of Thumb, what is the typical range of usual values?

$100 \pm 2 \times 2 = 100 \pm 4$, so 96$ cm to $104$ cm

Q: A dataset has $Q_1 = 20$, $Q_3 = 40$, and $IQR = 20$. What is the lower fence for detecting outliers?

$Q_1 - 1.5 \times IQR = 20 - 1.5 \times 20 = -10$

Q: A class of 20 students has test scores with a mean of 75 and a standard deviation of 5. A student scores 90. Is this score significantly high?

Yes, because $90 > 75 + 2 \times 5 = 85$

Question 1

A pharmaceutical company tests the lifespans (in years) of a new drug and finds the mean lifespan is 6 years with a standard deviation of 1.2 years. According to the Empirical Rule, what percentage of lifespans should fall between 3.6 and 8.4 years if the data is approximately bell-shaped?

Accepted Answer

$95\%$

Question 2

Which of the following measures of center is most resistant to the effect of outliers in a dataset of home prices in a city where a few homes are extremely expensive?

Accepted Answer

Median

Question 3

Given the following frequency distribution of commute times (in minutes):

Time: 0-10 (5), 10-20 (12), 20-30 (8), 30-40 (5)

What is the approximate mean commute time using class midpoints?

Accepted Answer

$\frac{5 	imes 5 + 12 	imes 15 + 8 	imes 25 + 5 	imes 35}{5+12+8+5} = 18.5$ minutes

Question 4

A student receives the following grades: 90 (weight 3), 80 (weight 2), 70 (weight 1). What is the weighted mean of the student's grades?

Accepted Answer

$\frac{90 	imes 3 + 80 	imes 2 + 70 	imes 1}{3+2+1} = 83.33$

Question 5

A dataset of exam scores is: 55, 60, 65, 70, 75, 80, 85. What is the interquartile range (IQR)?

Accepted Answer

$IQR = Q_3 - Q_1 = 80 - 60 = 20$

Question 6

A newborn weighs 4000 grams. If the mean birth weight is 3152 grams with a standard deviation of 693.4 grams, what is the newborn's $z$-score?

Accepted Answer

$z = \frac{4000 - 3152}{693.4} = 1.22$

Question 7

Which of the following statements about the standard deviation is TRUE?

Accepted Answer

The standard deviation measures the average distance of data values from the mean.

Question 8

A company produces metal rods with a mean length of 100 cm and a standard deviation of 2 cm. According to the Range Rule of Thumb, what is the typical range of usual values?

Accepted Answer

$100 \pm 2 	imes 2 = 100 \pm 4$, so 96$ cm to $104$ cm

Question 9

Given the dataset: 2, 4, 4, 4, 5, 5, 7, 9, what is the mode?

Accepted Answer

4

Question 10

A sample of 5 test scores is: 60, 70, 80, 90, 100. What is the sample variance?

Accepted Answer

$s^2 = \frac{(60-80)^2 + (70-80)^2 + (80-80)^2 + (90-80)^2 + (100-80)^2}{5-1} = 250$

Question 11

Which of the following is NOT part of the 5-number summary used in boxplots?

Accepted Answer

Mean

Question 12

A dataset has $Q_1 = 20$, $Q_3 = 40$, and $IQR = 20$. What is the lower fence for detecting outliers?

Accepted Answer

$Q_1 - 1.5 	imes IQR = 20 - 1.5 	imes 20 = -10$

Question 13

If a dataset is multimodal, which measure of center is most affected?

Accepted Answer

Mode

Question 14

A class of 20 students has test scores with a mean of 75 and a standard deviation of 5. A student scores 90. Is this score significantly high?

Accepted Answer

Yes, because $90 > 75 + 2 	imes 5 = 85$

Question 15

Which of the following best describes a percentile?

Accepted Answer

A value below which a given percentage of data falls

Question 16

A dataset has a mean of 50 and a standard deviation of 10. What is the $z$-score for a value of 70?

Accepted Answer

$z = \frac{70-50}{10} = 2$

Question 17

Which of the following is TRUE about the range as a measure of variation?

Accepted Answer

The range is not resistant to outliers.

Question 18

A dataset has the following 5-number summary: Min = 10, $Q_1 = 20$, Median = 30, $Q_3 = 40$, Max = 100. Which value is most likely an outlier?

Accepted Answer

100

Question 19

If a dataset has a mean of 60 and a standard deviation of 8, what is the approximate standard deviation using the Range Rule of Thumb if the minimum is 40 and the maximum is 80?

Accepted Answer

$s \approx \frac{80-40}{4} = 10$

Statistics Chapter 3: Measures of Center, Variation, and Relative Standing

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