Intro to Business Statistics: Random Variables and Probability Distributions - Normal Distribution Characteristics, Empirical Rule 6895997 ZScores Standard Normal Table

What This Is

The normal distribution is a fundamental concept in statistics that describes the distribution of many business variables, such as sales, stock prices, and quality control measurements. A retail chain wants to know if average daily sales exceed $10,000. By understanding the characteristics of the normal distribution, they can make informed decisions about inventory management, pricing, and marketing strategies.

Key Formulas & Symbols

• Z = (x? – ?) / (?/?n) where x? = sample mean,-= population mean,-= population standard deviation, n = sample size.
• Empirical Rule (68-95-99.7): About 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
• Standard Normal Table (Z-table): A table that shows the probability of observing a value less than or equal to a given Z-score.
• ? (Population Standard Deviation): A measure of the spread of the population distribution.
• (Sample Standard Deviation): An estimate of the population standard deviation based on sample data.
• x? (Sample Mean): The average value of a sample of data.
• ? (Population Mean): The average value of the population distribution.

Step-by-Step Procedure

1. State hypotheses: Clearly define the null and alternative hypotheses (e.g., H?:-= 10,000 vs. H?:-> 10,000).
2. Choose test: Select the appropriate statistical test (e.g., Z-test for normal distribution).
3. Compute test statistic: Calculate the Z-score using the formula Z = (x? – ?) / (?/?n).
4. Find p-value or critical value: Use the Z-table to find the probability of observing a value less than or equal to the calculated Z-score (p-value) or the critical value from the Z-table.
5. Compare to ?: Compare the p-value or critical value to the significance level (? = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H? if p-value < ?).

Common Mistakes

• Mistake: Using Z when-is unknown.
• Correction: Use the t-test instead, which is more robust when-is unknown.
• Mistake: Misinterpreting p-value as probability H? is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H? is true.
• Mistake: Failing to check assumptions (e.g., normality).
• Correction: Verify that the data meet the assumptions of the test (e.g., normal distribution).

Quick Practice Problems

1. A company claims that their average daily sales are $15,000. A sample of 36 days shows a mean of $14,500 with a standard deviation of $2,000. Is this claim plausible?

Answer: Z = (14,500 - 15,000) / (2,000 / ?36) = -1.25. Using the Z-table, the p-value is approximately 0.1056. Since p-value >-= 0.05, we fail to reject H?.

1. A quality control process claims that the average defect rate is 2%. A sample of 100 units shows a mean defect rate of 1.8% with a standard deviation of 0.5%. Is this claim plausible?

Answer: Z = (1.8 - 2) / (0.5 / ?100) = -1. The p-value is approximately 0.1587. Since p-value >-= 0.05, we fail to reject H?.

1. A marketing campaign claims that the average increase in sales is 10%. A sample of 25 stores shows a mean increase of 12% with a standard deviation of 5%. Is this claim plausible?

Answer: Z = (12 - 10) / (5 / ?25) = 1.2. Using the Z-table, the p-value is approximately 0.8849. Since p-value >-= 0.05, we fail to reject H?.

Last-Minute Cram Sheet

1. Z = (x? – ?) / (?/?n): Calculate Z-score using this formula.
2. 68-95-99.7: About 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
3. Z-table: Use this table to find p-value or critical value.
4. ? (Population Standard Deviation): A measure of the spread of the population distribution.
5. (Sample Standard Deviation): An estimate of the population standard deviation based on sample data.
6. x? (Sample Mean): The average value of a sample of data.
7. ? (Population Mean): The average value of the population distribution.
8. ? = 0.05: Default significance level.
9. p-value is NOT the probability that H? is true: It's the probability of observing the data (or more extreme) if H? is true.
10. Use t-test when-is unknown: More robust than Z-test when-is unknown.