Intro to Business Statistics: Hypothesis Testing OneSample tTest for Mean σ Unknown Degrees of Freedom

What This Is

The one-sample t-test for mean is a statistical method used to determine if the average value of a sample is significantly different from a known population mean. A retail chain wants to know if the average daily sales of its stores exceed $10,000. To answer this question, the chain collects data on daily sales from a random sample of its stores and uses a one-sample t-test to compare the sample mean to the known population mean of $10,000.

Key Formulas & Symbols

• t = (x̄ – μ) / (s / √n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size, and df = n - 1.
• t-distribution: a probability distribution used to calculate the probability of observing a t-statistic.
• degrees of freedom (df): the number of observations in the sample minus one (n - 1).
• p-value: the probability of observing a t-statistic at least as extreme as the one observed, assuming the null hypothesis is true.
• α: the significance level, set to 0.05 by default.
• H0: the null hypothesis, stating that the population mean is equal to a known value (e.g., μ = 10,000).
• H1: the alternative hypothesis, stating that the population mean is not equal to the known value (e.g., μ ≠ 10,000).

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses (H0 and H1).
2. Choose test: Select the one-sample t-test for mean since the population standard deviation is unknown.
3. Compute test statistic: Calculate the t-statistic using the formula t = (x̄ – μ) / (s / √n).
4. Find p-value or critical value: Use a t-distribution table or calculator to find the p-value or critical value for the calculated t-statistic and degrees of freedom (df).
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: If the p-value is less than α, reject the null hypothesis (H0); otherwise, fail to reject H0.

Common Mistakes

• Mistake: Using the Z-test when the population standard deviation is unknown.
• Correction: Use the t-test for mean since the population standard deviation is unknown.
• Mistake: Misinterpreting the p-value as the probability that H0 is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H0 is true.
• Mistake: Failing to check the assumptions of the t-test (normality and independence).
• Correction: Check the assumptions before conducting the t-test.

Quick Practice Problems

1. A company wants to know if the average price of its product is higher than $20. A random sample of 25 products has a mean price of $22.50 and a standard deviation of $3.50. What is the p-value?

Answer: 0.012 (The p-value is calculated using a t-distribution table or calculator with a t-statistic of 2.22 and 24 degrees of freedom.)

1. A marketing firm wants to know if the average response rate to its ads is higher than 5%. A random sample of 100 ads has a mean response rate of 6.2% and a standard deviation of 2.5%. What is the t-statistic?

Answer: 2.48 (The t-statistic is calculated using the formula t = (x̄ – μ) / (s / √n).)

1. A quality control team wants to know if the average defect rate of its products is lower than 2%. A random sample of 50 products has a mean defect rate of 1.8% and a standard deviation of 1.2%. What is the critical value for a two-tailed test with α = 0.05 and 49 degrees of freedom?

Answer: 2.01 (The critical value is found using a t-distribution table or calculator.)

Last-Minute Cram Sheet

1. t-test for mean: used to compare a sample mean to a known population mean when the population standard deviation is unknown.
2. t-distribution: a probability distribution used to calculate the probability of observing a t-statistic.
3. degrees of freedom (df): the number of observations in the sample minus one (n - 1).
4. p-value: the probability of observing a t-statistic at least as extreme as the one observed, assuming the null hypothesis is true.
5. α: the significance level, set to 0.05 by default.
6. H0: the null hypothesis, stating that the population mean is equal to a known value.
7. H1: the alternative hypothesis, stating that the population mean is not equal to the known value.
8. t-statistic: calculated using the formula t = (x̄ – μ) / (s / √n).
9. critical value: the t-value that separates the rejection region from the non-rejection region.
10. ⚠️ p-value is NOT the probability that H0 is true – it’s the probability of observing the data (or more extreme) if H0 is true.