test the claim that less than 30% of the students in class own a car at the 0.10 significance level.
To test the claim that less than 30% of the students in a class own a car at the 0.10 significance level, you would need to perform a hypothesis test. Here are the steps to carry out this test:
Step 1: Define the hypotheses:
- Null hypothesis (H₀): The proportion of students who own a car is equal to or greater than 30% (p ≥ 0.30)
- Alternative hypothesis (H₁): The proportion of students who own a car is less than 30% (p < 0.30)
Step 2: Choose the significance level (α):
The given significance level is 0.10, which means we're allowing for a 10% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 3: Collect the data:
Gather data by surveying a random sample of students from the class and determine how many of them own a car.
Step 4: Calculate the test statistic:
Compute the test statistic, which in this case would be a z-test statistic since we are dealing with proportions. The formula to calculate the test statistic for a proportion is:
z = (p̂ - p₀) / sqrt((p₀ * (1 - p₀)) / n)
- where p̂ is the sample proportion, p₀ is the hypothesized proportion (0.30 in this case), and n is the sample size.
Step 5: Determine the critical value:
To determine the critical value for this one-tailed test at the 0.10 significance level, you can refer to the standard normal distribution table or use statistical software. The critical value for a one-tailed test at a 0.10 significance level is approximately -1.28.
Step 6: Make a decision:
Compare the test statistic to the critical value. If the test statistic is less than the critical value, we reject the null hypothesis. If the test statistic is greater than or equal to the critical value, we fail to reject the null hypothesis.
Step 7: Conclusion:
Based on the decision made in step 6, draw a conclusion from the test results. If the null hypothesis is rejected, it suggests evidence in support of the alternative hypothesis. If the null hypothesis is not rejected, there is not enough evidence to support the alternative hypothesis.
Note: It is important to perform the calculations accurately and check if the assumptions for the test are met, such as the sample being random and independent, and the sample size being sufficiently large.