Suppose that we want to test the claim that the majority (> 50% ) of UOB students read newspapers on a daily basis. We sampled 70 students and found out that 42 of them read newspapers on a daily basis? Test the claim using á=0.10.
To test the claim that the majority of UOB students read newspapers on a daily basis, we can use a hypothesis test.
Step 1: Formulate the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The majority of UOB students do not read newspapers on a daily basis.
- Alternative hypothesis (Ha): The majority of UOB students read newspapers on a daily basis.
Step 2: Determine the test statistic to use:
Since we are testing a claim about proportions (the majority), we can use the one-sample proportion z-test.
Step 3: Set the significance level (α):
In this case, α = 0.10, which means we are willing to accept a 10% chance of committing a Type I error (rejecting the null hypothesis when it is true).
Step 4: Collect the data and calculate the test statistic:
Out of 70 sampled students, 42 of them read newspapers on a daily basis. To calculate the test statistic, we need to find the sample proportion (p̂) and the standard error (SE).
Sample proportion (p̂) = number of successes / sample size
p̂ = 42/70 = 0.6
Standard error (SE) = √[(p̂ * (1 - p̂))/n]
SE = √[(0.6 * 0.4)/70] ≈ 0.0621
Step 5: Calculate the test statistic (z-value):
The test statistic (z-value) can be calculated using the formula:
z = (p̂ - P) / SE
In this case, since the null hypothesis assumes that the majority is not above 50%, we can use P = 0.5.
z = (0.6 - 0.5) / 0.0621 ≈ 1.61
Step 6: Determine the critical value:
Since α = 0.10, we need to find the z-value that corresponds to a cumulative probability of 1 - α = 0.90. Using a standard normal distribution table or a calculator, we can find the critical value of z to be approximately 1.645.
Step 7: Compare the test statistic (z-value) with the critical value:
If the test statistic is within the critical region (greater than the critical value), we reject the null hypothesis. Otherwise, if the test statistic is not within the critical region, we fail to reject the null hypothesis.
In this case, the test statistic (z = 1.61) is less than the critical value (z = 1.645). Therefore, we fail to reject the null hypothesis.
Step 8: Interpret the results:
Based on the sample data and the hypothesis test, there is not enough evidence to conclude that the majority (>50%) of UOB students read newspapers on a daily basis. Keep in mind that failing to reject the null hypothesis does not prove the alternative hypothesis to be false, but rather suggests that there is insufficient evidence to support the alternative hypothesis.