Fewer young people are driving. In 1983, 87 percent of 19 year-olds had a driver's license. Twenty-five years later that percentage had dropped to 75 percent (University of Michigan Transportation Research Institute website, April 7, 2012). Suppose these results are based on a random sample of 1200 nineteen year-olds in 1983 and again in 2008.
To analyze the given data, we need to perform a hypothesis test for comparing proportions. Specifically, we want to compare the proportion of 19-year-olds with a driver's license in 1983 to the proportion in 2008.
Step 1: State the hypotheses:
In this case, our null hypothesis (H0) and alternative hypothesis (Ha) can be stated as follows:
H0: The proportion of 19-year-olds with a driver's license in 1983 is equal to the proportion in 2008.
Ha: The proportion of 19-year-olds with a driver's license in 1983 is different from the proportion in 2008.
Step 2: Set the significance level (α):
The significance level (α) determines the threshold for rejecting the null hypothesis. Here, we will assume a significance level of 0.05, which is a common choice.
Step 3: Collect and summarize the data:
According to the given information, we have two samples: 1200 nineteen year-olds in 1983 and 1200 nineteen year-olds in 2008. For each sample, we know the proportions of those who have a driver's license (87% in 1983 and 75% in 2008).
Step 4: Compute the test statistic:
To compare the proportions, we will use the formula for the test statistic for two proportions:
test statistic = (p1 - p2) / √((p̂ * (1 - p̂) / n1) + (p̂ * (1 - p̂) / n2))
Where:
p1 and p2 are the sample proportions,
p̂ is the pooled proportion ((p1*n1 + p2*n2) / (n1 + n2)),
n1 and n2 are the sample sizes.
In this case, p1 = 0.87, n1 = 1200, p2 = 0.75, n2 = 1200.
Step 5: Determine the critical value:
The critical value(s) depend on the significance level (α) and the specific test being used. As we are using a two-tailed hypothesis test, we will divide the significance level by 2 (0.025 on each tail) and find the corresponding z-scores from the standard normal distribution.
Step 6: Compute the p-value:
Using the test statistic and assuming a normal distribution (which is valid for large enough sample sizes), we can calculate the p-value associated with the observed test statistic. The p-value represents the probability of observing a test statistic as extreme or more extreme than the one obtained, assuming the null hypothesis is true.
Step 7: Make a decision:
Compare the p-value to the significance level (α) to decide whether to reject or fail to reject the null hypothesis. If the p-value is less than α, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Note: The calculations in steps 4-6 can be easily performed using statistical software or online calculators specifically designed for hypothesis testing.
With these steps, you can analyze the given data and determine whether there is a statistically significant difference in the proportion of 19-year-olds with a driver's license between 1983 and 2008.
To analyze the decrease in the percentage of 19-year-olds with a driver's license from 1983 to 2008, we can use a hypothesis test. The null hypothesis states that there is no difference in the proportion of 19-year-olds with a driver's license between the two years, while the alternative hypothesis states that there is a difference.
Let's define the proportion of 19-year-olds with a driver's license in 1983 as p1 and in 2008 as p2. We can compare these proportions using a two-sample proportion hypothesis test.
The given data provides the sample sizes for each year: 1200 nineteen year-olds in 1983 and 1200 nineteen year-olds in 2008.
To perform the hypothesis test, we need to calculate the sample proportions and the standard error of the difference.
Step 1: Calculate the sample proportions:
- The proportion of 19-year-olds with a driver's license in 1983 is 87% or 0.87, so p1 = 0.87.
- The proportion of 19-year-olds with a driver's license in 2008 is 75% or 0.75, so p2 = 0.75.
Step 2: Calculate the standard error (SE) of the difference:
- SE = sqrt [ (p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2) ]
- Where n1 = sample size in 1983 = 1200
- n2 = sample size in 2008 = 1200
Substituting the values:
SE = sqrt [ (0.87 * (1 - 0.87) / 1200) + (0.75 * (1 - 0.75) / 1200) ]
Step 3: Conduct the hypothesis test:
- Assuming a significance level (α) of 0.05, we will perform a two-tailed test.
- The critical z-score for a 95% confidence interval is approximately ± 1.96.
The formula for calculating the test statistic (z-score) is:
z = (p1 - p2) / SE
Substituting the values:
z = (0.87 - 0.75) / SE
Step 4: Compare the test statistic to the critical z-score:
- If the test statistic falls inside the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 5: Interpret the results:
- If we reject the null hypothesis, it means there is a significant difference in the proportion of 19-year-olds with a driver's license between 1983 and 2008.
Please note that without the actual data, we cannot compute the exact values. However, you can use this step-by-step procedure to perform the hypothesis test once you have the necessary information.