from generation to generation, the average age when smokers first start to smoke caties. however, the standatd deviation of that age remains constant of around 2.1 years. a survey of 40 smokers of this generation was done to see if the average starting age is at least 19. the sample average was 18.1 with a samole standatd deviation of 1.3.
To determine whether the average starting age for smokers of this generation is at least 19, we can perform a hypothesis test using the sample data provided. Let's follow the steps to conduct this hypothesis test:
Step 1: State the hypotheses.
- Null hypothesis (H0): The average starting age for smokers of this generation is 19 or less (μ <= 19).
- Alternative hypothesis (Ha): The average starting age for smokers of this generation is greater than 19 (μ > 19).
Step 2: Set the significance level (α).
- The significance level (α) is a predetermined threshold to determine the level of evidence required to reject the null hypothesis. Let's assume α = 0.05 (5%) for this test.
Step 3: Compute the test statistic.
- We'll use the t-test for this hypothesis test since the population standard deviation is not known.
- The formula for the t-test statistic is:
t = (x̄ - μ0) / (s / √n)
Where x̄ is the sample mean, μ0 is the hypothesized population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size.
- Given data:
x̄ (sample mean) = 18.1
μ0 (hypothesized mean) = 19
s (sample standard deviation) = 1.3
n (sample size) = 40
Substituting these values into the formula, we have:
t = (18.1 - 19) / (1.3 / √40)
Calculate the value of t.
Step 4: Determine the critical value.
- The critical value is the value beyond which we reject the null hypothesis.
- We will use the t-distribution with n - 1 degrees of freedom (df = 40 - 1 = 39) and the significance level α = 0.05.
- Find the critical value for a one-tailed test (right-tailed) at α = 0.05 and df = 39.
Step 5: Compare the test statistic with the critical value.
- If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Draw a conclusion.
- Based on the comparison in Step 5, we will draw a conclusion about the null hypothesis.
Now, let's perform the calculations to determine the test statistic and critical value.