A researcher wants to determine if two brands of cigarettes contain the same amount of nicotine. The data is shown in the table. With a significance level of 0.05 to that conclusion can be reached. Set the interval for a confidence of 95%
Sample A: 40.3 37.5 37 41.7 29.8 33.5
31.3 33.9 34.7 30.3 39.8 40.9 32.7 39.2
38.4 25.1 32.2 38 32.8 31.8
Sample B: 46.9 47.6 41.6 41.3 36.5 39.8
45.3 43.1 42.1 36.8 46.4 41.2 47.3 45.5
34.9 41.0 37.3 42.7 38.0 35.7 34.6 38.0
41.0 41.6 Assume That the two Populations Are Normally Distributed.
To determine if two brands of cigarettes contain the same amount of nicotine, you can perform a two-sample t-test.
Step 1: Formulate the hypotheses
- Null Hypothesis (H0): The mean nicotine content in brands A and B are equal.
- Alternative Hypothesis (Ha): The mean nicotine content in brands A and B are not equal.
Step 2: Calculate the sample statistics
- Sample A:
- Sample mean (x̄₁) = sum of all values in sample A / number of observations in sample A = (40.3 + ... + 32.8 + 31.8) / 20
- Sample standard deviation (s₁) = square root of [(sum of squared deviations from the mean) / (number of observations - 1)]
- Sample B:
- Sample mean (x̄₂) = sum of all values in sample B / number of observations in sample B = (46.9 + ... + 34.6 + 38.0) / 23
- Sample standard deviation (s₂) = square root of [(sum of squared deviations from the mean) / (number of observations - 1)]
Step 3: Determine the critical value
- The critical value is based on the significance level (0.05) and the degrees of freedom (df), which is the sum of the sample sizes minus 2 (df = n₁ + n₂ - 2). You can find the critical value using a t-distribution table or statistical software.
Step 4: Perform the t-test
- Calculate the t-statistic using the formula:
t = (x̄₁ - x̄₂) / sqrt((s₁² / n₁) + (s₂² / n₂))
- Compare the t-statistic with the critical value. If the absolute value of the t-statistic is greater than the critical value, reject the null hypothesis.
Step 5: Determine the confidence interval
- To find the confidence interval, use the formula:
Confidence interval = (x̄₁ - x̄₂) ± t * sqrt((s₁² / n₁) + (s₂² / n₂)). Here, t is the critical value for the desired confidence level.
For a confidence level of 95%, use a critical value for t (two-tailed test) with degrees of freedom equal to n₁ + n₂ - 2.
Perform these calculations using the given data and follow the steps to reach your conclusion on whether the mean nicotine content in brands A and B are equal.