A research center claims that 50% of people believe that drivers should be allowed to use cellular phones with hands free devices while driving. In a random sample of 150 U.S. adults 58% say that drivers should be allowed to use cellular phones with hands free devices while driving. At a=0.01, is there enough evidence to reject the centers claim?
State null, alternative, identify claim
Fine critical and critical regions
Compute test
make decision.
You can use a one-sample proportional z-test for your data. (Test sample proportion = .58 and sample size = 150) Find the critical value in the appropriate table at .01 level of significance for a two-tailed test. Compare the test statistic you calculate to the critical value from the table. If the test statistic exceeds the critical value, reject the null. If the test statistic does not exceed the critical value, do not reject the null. You can draw your conclusions from there.
To determine if there is enough evidence to reject the research center's claim, we need to conduct a hypothesis test. Here's how we can approach this problem:
1. State the null and alternative hypotheses:
- Null hypothesis (H0): The true proportion of people who believe drivers should be allowed to use cellular phones with hands free devices while driving is 0.50.
- Alternative hypothesis (Ha): The true proportion of people who believe drivers should be allowed to use cellular phones with hands free devices while driving is not equal to 0.50.
2. Identify the claim:
- The claim from the research center is that 50% of people believe drivers should be allowed to use cellular phones with hands free devices while driving.
3. Define the significance level:
- The given significance level is α=0.01. This determines how extreme the evidence must be for us to reject the null hypothesis.
4. Determine the critical and critical regions:
- In this case, the hypothesis test is a two-tailed test because the alternative hypothesis is not equal to 0.50. Therefore, we will divide the significance level (α) by 2 to get the critical regions.
- α/2 = 0.01/2 = 0.005 (for each tail)
- The critical regions are the two extreme areas of the sampling distribution, which fall beyond ±0.005 from the expected proportion of 0.50.
5. Compute the test statistic:
- The test statistic we need to compute is the z-score, which measures how far the sample proportion is from the expected population proportion.
- The formula for calculating the z-score is: z = (p̂ - p) / √[(p * (1 - p))/n], where p̂ is the sample proportion, p is the expected population proportion, and n is the sample size.
- In this case, p̂ = 0.58 (from the sample), p = 0.50 (from the null hypothesis), and n = 150 (sample size).
6. Make a decision:
- To make a decision, we compare the test statistic (z-score) to the critical values obtained in step 4.
- If the test statistic falls within the critical region(s), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
So, to determine if there is enough evidence to reject the research center's claim, we need to calculate the z-score and compare it to the critical values.