This Discussion Board requires you to answer four questions. For each question, do not simply provide an answer; make sure you explain how you arrived at that answer.
Questions
Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim. Superior Motor Company claims that its new sedan, the Astro, will average better than 42 miles per gallon on the highway. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.
Provide an appropriate response. Tim believes that a coin is coming up tails less than 50% of the time. He tests the claim p < 0.5. In 100 tosses, the coin comes up tails 48 times. What is the value of the sample proportion? Do you think the P-value will be small or large and what should Tim conclude about the claim p < 0.5?
Based on a simple random sample of students from her school, Sally obtained a point estimate of the mean weight of students at her school. What additional information would be provided by a confidence interval estimate of the mean weight? Provide a supporting example.
Assume that a simple random sample has been selected from a normally distributed population and test the given claim:
Once a Crime Scene Specialist is called to respond to a Crime Scene (homicide, suicide, sexual assault, etc.) the mean response time should be no more than 10 minutes if within city limits. For a local police department, 18 different call outs and their associated response times were documented. The mean response time was 7.6 minutes with a standard deviation of 2.3 minutes. At the 0.01 significance level, test the claim that the mean response time is less than 10 minutes.
Identify the null and alternative hypotheses
Calculate the test statistic
Find the critical value and associated P-value
State the final conclusion that addresses the original claim
Formulate the indicated conclusion in nontechnical terms:
Superior Motor Company claimed that its new sedan, the Astro, would achieve better than 42 miles per gallon on the highway. After conducting a hypothesis test, if the conclusion is to reject the null hypothesis, this means that there is enough evidence to support the claim that the Astro sedan does indeed average better than 42 miles per gallon on the highway.
Response to Tim's claim about coin tosses:
In Tim's case, he believed that a coin would come up tails less than 50% of the time. After conducting 100 tosses and observing 48 tails, the sample proportion would be calculated as 48/100 = 0.48. Since the observed proportion is close to 0.5, the P-value is likely to be large, indicating that there is not enough evidence to conclude that the coin comes up tails less than 50% of the time.
Additional information from a confidence interval estimate:
Sally obtained a point estimate of the mean weight of students at her school from a sample. By calculating a confidence interval for the mean weight, she would gain additional information about how confident she can be that the true mean weight of all students falls within a certain range. For example, if Sally calculated a 95% confidence interval for the mean weight of students at her school as 140 lbs to 160 lbs, she would be 95% confident that the true mean weight lies within this interval.
Testing the claim about crime scene response time:
- Null hypothesis: The mean response time is 10 minutes.
- Alternative hypothesis: The mean response time is less than 10 minutes.
- Test statistic: t = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size)) = (7.6 - 10) / (2.3 / sqrt(18)) ≈ -3.08
- Critical value and P-value: With a significance level of 0.01 and degrees of freedom (df) = sample size - 1 = 18 - 1 = 17, the critical value can be found from a t-distribution table or calculator. The P-value can also be determined based on the test statistic and degrees of freedom.
- Conclusion: If the calculated P-value is less than 0.01,
there is enough evidence to reject the null hypothesis and support the claim that the mean response time is less than 10 minutes within city limits based on the sample data collected.