Intro to Business Statistics: Correlation and Regression - Hypothesis Testing for Slope, t-Test and Overall FTest

What This Is

Hypothesis testing for slope (t-test) and overall F-test are statistical methods used to determine if there's a significant relationship between two variables in a linear regression model. A retail chain wants to know if the average daily sales exceed $10,000 when the price of an item is below $20. They collect data on sales and prices for 25 days and want to test if the slope of the regression line is significantly different from zero.

Key Formulas & Symbols

• t = (b - ?) / (s_b / ?(1 + 1/n)) where b = sample slope,-= population slope, s_b = sample standard error of the slope, n = sample size.
• t = test statistic
• b = sample slope
• ? = population slope
• s_b = sample standard error of the slope
• n = sample size
• F = (MSR / MSE) where MSR = mean square regression, MSE = mean square error.
• F = F-statistic
• MSR = mean square regression
• MSE = mean square error
• p-value = P(T-|t|) where T = t-distribution with n-2 degrees of freedom.
• p-value = probability of observing the data (or more extreme) if H? is true
• T = t-distribution
• n = sample size
• t-critical = t_(n-2, ?/2) where n = sample size,-= significance level.
• t-critical = critical value from t-distribution
• n = sample size
• ? = significance level
• F-critical = F_(n-2, n-p-1, ?) where n = sample size, p = number of predictors,-= significance level.
• F-critical = critical value from F-distribution
• n = sample size
• p = number of predictors
• ? = significance level
• H?:-= 0 where-= population slope.
• H? = null hypothesis
• ? = population slope
• H_a:-? 0 where-= population slope.
• H_a = alternative hypothesis
• ? = population slope

Step-by-Step Procedure

1. State hypotheses: Write H? and H_a based on the research question.
2. Choose test: Select the t-test for slope or F-test for overall model significance.
3. Compute test statistic: Calculate t or F using the formulas above.
4. Find p-value or critical value: Determine the p-value or critical value from the t or F distribution.
5. Compare to ?: Compare the p-value or critical value to the significance level (? = 0.05).
6. Conclude: Reject H? if p-value <-or F-statistic > F-critical.

Common Mistakes

• Mistake: Using Z when-is unknown.
• Correction: Use t-distribution when-is unknown, as it's a more robust test.
• Mistake: Misinterpreting p-value as probability H? is true.
• Correction: p-value is the probability of observing the data (or more extreme) if H? is true.
• Mistake: Failing to check assumptions (linearity, independence, normality).
• Correction: Check assumptions before conducting the test to ensure validity.

Quick Practice Problems

1. A marketing firm wants to know if the average daily sales exceed $10,000 when the price of an item is below $20. They collect data on sales and prices for 25 days and want to test if the slope of the regression line is significantly different from zero. Calculate the t-statistic.

t = (b - ?) / (s_b / ?(1 + 1/n)) = (0.5 - 0) / (0.2 / ?(1 + 1/25)) = 2.5

1. A retail chain wants to know if the overall model is significant. They collect data on sales and prices for 30 days and want to test if the F-statistic is greater than the F-critical value. What is the p-value?

p-value = P(F-3.3) = 0.01

1. A company wants to know if the slope of the regression line is significantly different from zero. They collect data on sales and prices for 20 days and want to test if the t-statistic is greater than the t-critical value. What is the decision?

Reject H? since t = 2.1 > t-critical = 1.7

Last-Minute Cram Sheet

1. t-test for slope: Use when testing a single slope coefficient.
2. F-test for overall model: Use when testing the overall significance of the model.
3. p-value: Probability of observing the data (or more extreme) if H? is true.
4. t-distribution: Use when-is unknown and sample size is small.
5. F-distribution: Use when testing the overall significance of the model.
6. ? = 0.05: Default significance level.
7. H?:-= 0: Null hypothesis for slope.
8. H_a:-? 0: Alternative hypothesis for slope.
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-distribution when-is unknown, as it’s a more robust test.