Q1. Calculate the Measures of Central Tendency and Dispersion for the sample data: 10, 12, 15, 22, 22, 34, 42. (Round to two decimal places.)

Background

Topic: Descriptive Statistics

This question tests your ability to compute key statistics: mean, median, mode, range, variance, and standard deviation for a given data set.

Key Terms and Formulas:

• Mean (Sample):

• Median: Middle value when data is ordered

• Mode: Most frequent value

• Range:

• Variance (Sample):

• Standard Deviation (Sample):

Step-by-Step Guidance

1. Order the data: 10, 12, 15, 22, 22, 34, 42.

2. Count the number of data points ().

3. Calculate the mean:

4. Find the median: Since is odd, the median is the 4th value in the ordered list.

5. Identify the mode: Look for the value(s) that appear most frequently.

6. Compute the range:

7. Calculate variance: First, find for each value, square each, sum them, then divide by .

8. Standard deviation: Take the square root of the variance.

Try solving on your own before revealing the answer!

Q2a. Write the formulas for the mean for a sample AND for a population, using proper notation.

Background

Topic: Measures of Central Tendency

This question checks your understanding of the difference between sample and population mean formulas.

Key Terms and Formulas:

• Sample mean:

• Population mean:

Step-by-Step Guidance

1. Recall that is used for sample mean, for population mean.

2. Write the formula for sample mean using (sample size).

3. Write the formula for population mean using (population size).

Try writing the formulas before checking the answer!

Q2b. Write the formulas for the standard deviation for a sample AND for a population, using proper notation.

Background

Topic: Measures of Dispersion

This question tests your ability to distinguish between sample and population standard deviation formulas.

Key Terms and Formulas:

• Sample standard deviation:

• Population standard deviation:

Step-by-Step Guidance

1. Recall that is for sample, for population.

2. Write the formula for sample standard deviation using in the denominator.

3. Write the formula for population standard deviation using in the denominator.

Try writing the formulas before checking the answer!

Q2c. Sketch the three approximate percentages and their corresponding number of standard deviations for the Empirical Rule.

Background

Topic: Empirical Rule (68-95-99.7 Rule)

This question tests your understanding of how data is distributed in a normal distribution and the percentages within 1, 2, and 3 standard deviations from the mean.

Key Terms:

• Empirical Rule: In a normal distribution, about 68% of data falls within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

Step-by-Step Guidance

1. Draw a bell-shaped curve (normal distribution).

2. Mark the mean () at the center.

3. Label , , .

4. Write the percentages: 68% between , 95% between , 99.7% between .

Try sketching and labeling the curve before checking the answer!

Q3. Data are drawn from a normal distribution with mean 40, standard deviation 4. Use the Empirical Rule. SKETCH the values and calculate the Z scores.

Background

Topic: Normal Distribution, Empirical Rule, Z Scores

This question tests your ability to apply the Empirical Rule and calculate Z scores for given values.

Key Terms and Formulas:

• Z score:

• Empirical Rule: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

Step-by-Step Guidance

1. Identify mean () and standard deviation ().

2. For each sub-question, determine the relevant interval in terms of SDs from the mean.

3. Calculate Z scores for the endpoints using .

4. Use the Empirical Rule to estimate the percentage of observations in each interval.

Try calculating the Z scores and percentages before checking the answer!

Q4. The following data show the miles per gallon of small SUVs: 39, 31, 39, 35, 39, 29, 40, 43, 44, 44, 47. Sketch a boxplot and calculate: Smallest, Range, Q1, IQR, Median, Outliers (show calculation), Q3, Largest.

Background

Topic: Boxplots and Five-Number Summary

This question tests your ability to construct a boxplot, calculate quartiles, interquartile range, and identify outliers.

Key Terms and Formulas:

• Five-number summary: Min, Q1, Median, Q3, Max

• Interquartile Range (IQR):

• Outlier criteria: Values below or above

Step-by-Step Guidance

1. Order the data from smallest to largest.

2. Identify the smallest and largest values.

3. Find the median (middle value).

4. Calculate Q1 (lower quartile) and Q3 (upper quartile).

5. Compute IQR:

6. Check for outliers using the outlier criteria.

Try calculating the quartiles and IQR before checking the answer!

Q5. The monthly apartment rental rates near Enterprise College approximate a symmetrical, bell-shaped distribution. The sample mean rental rate is $800; the standard deviation is $50. Use the Empirical Rule.

Background

Topic: Normal Distribution, Empirical Rule, Z Scores

This question tests your ability to apply the Empirical Rule to real-world data and calculate Z scores.

Key Terms and Formulas:

• Empirical Rule: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

• Z score:

Step-by-Step Guidance

1. Identify mean () and standard deviation ().

2. For part (a), calculate the interval .

3. For part (b), determine how many SDs $750 are from the mean, then use the Empirical Rule.

4. For part (c), calculate the Z score for $730z = \frac{730 - 800}{50}$.

Try calculating the intervals and Z scores before checking the answer!

Q6. y = 62.65 – 1.86x1 – 0.52x2. y is monthly natural gas consumption in cubic feet; x1 is thickness of insulation (inches); x2 is outdoor temperature (F).

Background

Topic: Multiple Linear Regression

This question tests your ability to interpret and use a regression equation to make predictions and understand the meaning of coefficients.

Key Terms and Formulas:

• Regression equation:

• Interpretation: and are the change in per unit increase in and respectively.

Step-by-Step Guidance

1. For part (a), substitute , into the equation.

2. Calculate each term: and .

3. Add these to to get the expected value.

4. For part (b), compare the result for to (with ).

5. For part (c), explain why negative coefficients make sense in this context.

Try substituting the values and interpreting the coefficients before checking the answer!

Q7. Select a committee of 3 people from your staff of 12. How many ways can this be accomplished when one person will be the lead, one will be the record keeper, one will be the scheduler?

Background

Topic: Counting Principles (Permutations)

This question tests your understanding of permutations when assigning distinct roles to committee members.

Key Terms and Formulas:

• Permutation formula:

• For distinct roles, order matters.

Step-by-Step Guidance

1. First, select 3 people from 12. Since roles are distinct, order matters.

2. Calculate the number of ways to assign the roles: (lead, record keeper, scheduler).

3. Alternatively, use the permutation formula:

Try calculating the number of ways before checking the answer!