Business Statistics Exam Study Guide: Step-by-Step Guidance

Q1. Data: 8, 12, 15, 10, 5. Find the mean, variance, and standard deviation.

Background

Topic: Descriptive Statistics – Measures of Center and Spread

This question tests your ability to compute the sample mean, variance, and standard deviation for a small data set. These are foundational concepts in statistics for summarizing and understanding data.

Key Terms and Formulas

• Sample Mean: $\bar{x} = \frac{\sum x}{n}$

• Sample Variance: $s^2 = \frac{\sum (x-\bar{x})^2}{n-1}$

• Sample Standard Deviation: $s = \sqrt{s^2}$

Step-by-Step Guidance

1. Add all the data values together to find the sum $\sum x$.

2. Count the number of data points $n$.

3. Calculate the mean using $\bar{x} = \frac{\sum x}{n}$.

4. Subtract the mean from each data value, square the result, and sum these squared differences to get $\sum (x-\bar{x})^2$.

5. Divide this sum by $n-1$ to find the sample variance $s^2$.

6. Take the square root of the variance to find the standard deviation $s$.

Try solving on your own before revealing the answer!

Final Answers:

Mean: 10

Variance: 14.5

Standard deviation: 3.81

Each step follows the formulas above. Calculating the mean gives the central value, variance measures average squared deviation, and standard deviation is the square root of variance.

Q2. Given mean = 40, SD = 8, X = 56. Find the z-score.

Background

Topic: Standardization (Z-Score)

This question tests your ability to standardize a value using the z-score formula, which tells you how many standard deviations a value is from the mean.

Key Terms and Formula

• Z-score: $z = \frac{X - \mu}{\sigma}$

• $X$ = observed value, $\mu$ = mean, $\sigma$ = standard deviation

Step-by-Step Guidance

1. Identify the observed value $X = 56$, mean $\mu = 40$, and standard deviation $\sigma = 8$.

2. Subtract the mean from the observed value: $X - \mu$.

3. Divide the result by the standard deviation: $z = \frac{X - \mu}{\sigma}$.

Try solving on your own before revealing the answer!

Final Answer: z = 2

Plugging in the values: $z = \frac{56 - 40}{8} = 2$

This means the value is 2 standard deviations above the mean.

Q3. Q1 = 22, Q3 = 50. Find the interquartile range (IQR).

Background

Topic: Measures of Spread – Interquartile Range

This question tests your understanding of quartiles and how to calculate the interquartile range, which measures the spread of the middle 50% of data.

Key Terms and Formula

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• $Q_1$ = first quartile (25th percentile), $Q_3$ = third quartile (75th percentile)

Step-by-Step Guidance

1. Identify $Q_1$ and $Q_3$ from the problem: $Q_1 = 22$, $Q_3 = 50$.

2. Subtract $Q_1$ from $Q_3$ to find the IQR: $IQR = Q_3 - Q_1$.

Try solving on your own before revealing the answer!

Final Answer: IQR = 28

$IQR = 50 - 22 = 28$

The IQR shows the range of the middle half of the data.

Q4. Mean = 100, SD = 20. Find the coefficient of variation (CV).

Background

Topic: Relative Measures of Spread

This question tests your ability to compute the coefficient of variation, which expresses the standard deviation as a percentage of the mean.

Key Terms and Formula

• Coefficient of Variation: $CV = \frac{s}{\bar{x}}$

• Often expressed as a percentage: $CV \times 100\%$

Step-by-Step Guidance

1. Identify the standard deviation $s = 20$ and mean $\bar{x} = 100$.

2. Divide the standard deviation by the mean: $CV = \frac{20}{100}$.

3. If asked for a percentage, multiply the result by 100.

Try solving on your own before revealing the answer!

Final Answer: CV = 0.2 or 20%

$CV = \frac{20}{100} = 0.2$ or $20\%$

This means the standard deviation is 20% of the mean.

Q5. r = -0.78. Interpret the strength and direction of the correlation.

Background

Topic: Correlation

This question tests your understanding of the correlation coefficient, which measures the strength and direction of a linear relationship between two variables.

Key Terms

• Correlation coefficient ($r$): ranges from -1 to 1

• Strength: closer to -1 or 1 means stronger relationship

• Direction: positive ($r > 0$) or negative ($r < 0$)

Step-by-Step Guidance

1. Look at the sign of $r$. Negative means the relationship is negative (as one variable increases, the other decreases).

2. Assess the magnitude. $|r| = 0.78$ is close to 1, indicating a strong relationship.

3. Combine both to interpret: strong negative linear relationship.

Try interpreting on your own before revealing the answer!

Final Answer:

There is a strong negative linear relationship between the two variables.

The negative sign means as one increases, the other tends to decrease.