Q1. Calculate the Measures of Central Tendency and Dispersion for the given sample data.

Background

Topic: Measures of Central Tendency and Dispersion

This question tests your ability to compute the mean, median, mode, range, variance, and standard deviation for a set of sample data.

Key Terms and Formulas:

• Mean ():

• Median: The middle value when data is ordered

• Mode: The value that appears most frequently

• Range:

• Variance (sample):

• Standard Deviation (sample):

Step-by-Step Guidance

1. Order the data from smallest to largest to help find the median and mode.

2. Calculate the mean by summing all values and dividing by the number of data points.

3. Find the median by locating the middle value (or averaging the two middle values if the sample size is even).

4. Identify the mode by determining which value occurs most frequently.

5. Compute the range by subtracting the smallest value from the largest value.

6. Set up the formula for variance and standard deviation, but stop before plugging in the final values.

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Q2. Write the formulas for the mean and standard deviation for a sample AND for a population, using proper notation.

Background

Topic: Statistical Notation and Formulas

This question tests your understanding of the difference between sample and population statistics, and your ability to use correct notation.

Key Terms and Formulas:

• Sample mean:

• Population mean:

• Sample standard deviation:

• Population standard deviation:

Step-by-Step Guidance

1. Write out the formula for the mean for both sample and population, noting the difference in notation ( vs ).

2. Write out the formula for standard deviation for both sample and population, noting the difference in denominator ( vs ).

3. Explain why the sample formula uses (Bessel's correction) and the population formula uses .

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Q3. Use the Empirical Rule to estimate percentages and corresponding number of standard deviations for a normal distribution.

Background

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

This question tests your understanding of the Empirical Rule, which describes the percentage of data within 1, 2, and 3 standard deviations of the mean in a normal distribution.

Key Terms and Formulas:

• Empirical Rule: About 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.

• Standard deviation (): Measure of spread in a normal distribution.

Step-by-Step Guidance

1. Draw or visualize a normal distribution curve.

2. Mark the mean and 1, 2, and 3 standard deviations from the mean.

3. Label the percentage of data within each interval according to the Empirical Rule.

4. Explain how to use these percentages to estimate probabilities or counts in a data set.

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Q4. Data are drawn from a normal distribution with a mean of 40 and standard deviation of 6. Use the Empirical Rule to answer questions about percentages and Z-scores.

Background

Topic: Normal Distribution, Z-scores, Empirical Rule

This question tests your ability to use the Empirical Rule and Z-scores to estimate probabilities and percentages for values in a normal distribution.

Key Terms and Formulas:

• Z-score:

• Empirical Rule: See above.

Step-by-Step Guidance

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

2. Calculate Z-scores for the values in question using .

3. Use the Empirical Rule to estimate the percentage of observations within the specified Z-score range.

4. Shade or mark the relevant area under the normal curve to visualize the percentage.

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Q5. Calculate measures of central tendency and identify outliers for a data set showing miles per gallon of small SUVs.

Background

Topic: Measures of Central Tendency, Outlier Identification

This question tests your ability to compute mean, median, mode, range, and identify outliers using the interquartile range (IQR) method.

Key Terms and Formulas:

• Mean, Median, Mode, Range: See above.

• Interquartile Range (IQR):

• Outlier: Any value less than or greater than

Step-by-Step Guidance

1. Order the data and calculate the mean, median, mode, and range.

2. Find the first quartile () and third quartile ().

3. Calculate the IQR and use it to determine the boundaries for outliers.

4. Identify any values outside these boundaries as outliers.

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Q6. Use regression equation to predict monthly natural gas consumption based on insulation thickness and outdoor temperature.

Background

Topic: Linear Regression

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

Key Terms and Formulas:

• Regression equation:

• : Predicted value (monthly gas consumption)

• : Thickness of insulation

• : Outdoor temperature

Step-by-Step Guidance

1. Identify the regression equation and the meaning of each variable and coefficient.

2. Plug in the given values for insulation thickness and outdoor temperature.

3. Set up the calculation for but stop before computing the final value.

4. Interpret the regression coefficients: explain how changes in and affect .

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