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.
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To determine whether there is enough evidence to reject the research center's claim, we can conduct a hypothesis test. Here's how you can approach it step by step:
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 equal to 50%.
- Alternative hypothesis (H₁): 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 50%.
2. Identify the claim:
The research center claims that 50% of people believe drivers should be allowed to use cellular phones with hands-free devices while driving.
3. Determine the critical and critical regions:
Since the significance level (α) is given as 0.01, we need to divide it by 2 for a two-tailed test. This gives us α/2 = 0.005. We will use this value to determine the critical regions.
4. Compute the test statistic:
The test statistic for hypothesis testing with proportions is the z-score. We can calculate it using the formula:
z = (p̂ - p₀) / √(p₀(1 - p₀) / n)
where p̂ is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.
In this case:
p̂ = 0.58 (sample proportion)
p₀ = 0.50 (hypothesized proportion)
n = 150 (sample size)
Plugging in the values, we get:
z = (0.58 - 0.50) / √(0.50(1 - 0.50) / 150)
5. Make a decision:
To make a decision, we compare the test statistic (z-score) with the critical values associated with the critical regions.
- If the z-score falls within the critical region (outside the critical values), we reject the null hypothesis.
- If the z-score does not fall within the critical region (inside the critical values), we fail to reject the null hypothesis.
To find the critical values, we can use a standard normal distribution table or a calculator. At α/2 = 0.005 significance level, the critical values are approximately ±2.576.
So, if the absolute value of the calculated z-score is greater than 2.576, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Now you can calculate the test statistic (z-score) and make a decision based on the information provided.