Statistics Practice: Empirical Rule, Z‑Scores & Regression Results

Statistics Problems and Solutions

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Question 1: Empirical Rule Name

The Empirical Rule is also known as:

• E. The 68–95–99.7 Rule

Question 2: Normal Distribution Interval (95%)

For a normally distributed data set with a mean of 70 and a standard deviation of 13, approximately 95% of the data lie within about two standard deviations of the mean.

• Calculate: 70 ± 2 × 13 = 70 ± 26
• Result: 44 and 96

Question 3: Mean and Median for the Set

For the set of numbers: 27, 83, 55, 61, 102, 17:

• a) Mean — Add all the numbers together and divide by the count (6):
• Sum = 27 + 83 + 55 + 61 + 102 + 17 = 345
• Mean = 345 / 6 = 57.5
• b) Median — Arrange in ascending order and take the middle value (or average the two middle values for even count):
• Sorted: 17, 27, 55, 61, 83, 102
• Median = (55 + 61) / 2 = 58

Question 4: Z‑Score for a 6‑Foot Woman

To calculate the z‑score for a woman 6 feet tall:

• Convert 6 feet to inches: 6 feet × 12 inches/foot = 72 inches
• Use the z‑score formula: z = (x - mean) / standard deviation
• Given mean = 65.5 and standard deviation = 2.5:
• z = (72 − 65.5) / 2.5 = 6.5 / 2.5 = 2.60

Question 5: Regression Variable Name

In a regression model, the variable x is called the independent variable.

Question 6: R‑Squared from Correlation

If the correlation coefficient is −0.8, R‑squared is the square of the correlation coefficient:

• R² = (−0.8)² = 0.64

Part B: Computational Questions

1) Standard Deviation for the Data Set

Data: −4, −2, 5, 7, 9, 13

• Step 1 — Calculate the mean:
• Mean = (−4 + (−2) + 5 + 7 + 9 + 13) / 6 = 28 / 6 = 4.666... (4.67)
• Step 2 — Compute squared deviations (xi − mean)² for each value:
• (−4 − 4.6667)² ≈ 75.1111
• (−2 − 4.6667)² ≈ 44.4444
• (5 − 4.6667)² ≈ 0.1111
• (7 − 4.6667)² ≈ 5.4444
• (9 − 4.6667)² ≈ 18.7778
• (13 − 4.6667)² ≈ 69.4444
• Step 3 — Sum of squared deviations ≈ 75.1111 + 44.4444 + 0.1111 + 5.4444 + 18.7778 + 69.4444 = 213.3333
• Step 4 — Divide by N (population standard deviation formula used here): 213.3333 / 6 = 35.5556
• Step 5 — Square root: sqrt(35.5556) ≈ 5.96
• Result: The population standard deviation is approximately 5.96.

2) Wheat Production and Flour Price Analysis

For the wheat production and flour price data:

• a) Compute the correlation coefficient, r — use the correlation formula to compute r from your paired data.
• b) Compute the equation of the regression model — use least squares regression to find the intercept and slope (ŷ = b0 + b1x).
• c) Compute the residuals — residuals = actual value − predicted value for each observation.
• d) Compute the standard error of the estimate, se — this measures the typical size of residuals and the accuracy of the model's predictions.

Notes: For items a–d, apply the standard formulas for correlation and linear regression to your paired wheat-production and flour-price data. Residuals are useful to check model fit; the standard error of the estimate is sqrt( sum(residuals²) / (n − 2) ) for simple linear regression (using n − 2 degrees of freedom).

If you would like, provide the actual wheat and flour-price data and I will compute r, the regression equation, the residuals, and the standard error step by step.