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Chapter 9: Problem 46

The article "First Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" (Journal of College Student Development \([1999]: 599-610\) ) reported that the sample mean and standard deviation for high school grade point average (GPA) for students enrolled at a large research university were \(3.73\) and \(0.45\), respectively. Suppose that the mean and standard deviation were based on a random sample of 900 students at the university. a. Construct a \(95 \%\) confidence interval for the mean high school GPA for students at this university. b. Suppose that you wanted to make a statement about the range of GPAs for students at this university. Is it reasonable to say that \(95 \%\) of the students at the university have GPAs in the interval you computed in Part (a)? Explain.

Short Answer

Expert verified
a. The 95% confidence interval for the mean high school GPA for students at this university is estimated to be between 3.70 and 3.76. b. It is not reasonable to say that 95% of the students at the university have GPAs in the interval computed in Part (a). This is because the calculated confidence interval reflects where we believe the true mean GPA lies, not the individual students' GPA.

Step by step solution

01

Calculate the Confidence Interval

First, calculate the standard error. The standard error (\( SE \)) is given by: \[ SE = \frac{{\text{{Standard deviation}}}}{{\sqrt{{\text{{Sample size}}}} }} \]where the standard deviation (SD) is \( 0.45 \) and the sample size (N) is 900. So the standard error is \[ SE = \frac{{0.45}}{{\sqrt{{900}} }} = 0.015 \]For a 95% confidence level, the Z-value is approximately 1.96 (You can find this value in Z-table or online).Now, calculate the confidence interval using the given formula:\[\text{{Confidence Interval}} = \text{{Sample mean}} \pm (z \times SE ) = 3.73 \pm (1.96 \times 0.015)\]This gives the 95% Confidence Interval as \( (3.73 - 0.0294 , 3.73 + 0.0294) = (3.70, 3.76) \).
02

Interpretation and Discussion of Part (b)

To answer part (b), it's crucial to understand that even though the Confidence Interval has been calculated to be 95%, the interpretation of this does not mean that 95% of all students have a GPA falling within this range. The Confidence Interval is a reflection of where we expect the true mean GPA of the entire student population to fall, given our sample data, not the individual GPAs. Hence it is not reasonable to state that 95% of the students at the university have GPAs in the interval computed in Part (a).

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Most popular questions from this chapter

The formula used to compute a large-sample confidence interval for \(\pi\) is $$ p \pm(z \text { critical value }) \sqrt{\frac{p(1-p)}{n}} $$ What is the appropriate \(z\) critical value for each of the following confidence levels? a. \(95 \%\) d. \(80 \%\) b. \(90 \%\) e. \(85 \%\) c. \(99 \%\)

The article "CSI Effect Has Juries Wanting More Evidence" (USA Today, August 5,2004 ) examines how the popularity of crime-scene investigation television shows is influencing jurors' expectations of what evidence should be produced at a trial. In a survey of 500 potential jurors, one study found that 350 were regular watchers of at least one crime-scene forensics television series. a. Assuming that it is reasonable to regard this sample of 500 potential jurors as representative of potential jurors in the United States, use the given information to construct and interpret a \(95 \%\) confidence interval for the true proportion of potential jurors who regularly watch at least one crime-scene investigation series. b. Would a \(99 \%\) confidence interval be wider or narrower than the \(95 \%\) confidence interval from Part (a)?

In an AP-AOL sports poll (Associated Press, December 18,2005 ), 394 of 1000 randomly selected U.S. adults indicated that they considered themselves to be baseball fans. Of the 394 baseball fans, 272 stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. a. Construct a \(95 \%\) confidence interval for the proportion of U.S. adults that consider themselves to be baseball fans. b. Construct a \(95 \%\) confidence interval for the proportion of those who consider themselves to be baseball fans that think the designated hitter rule should be expanded to both leagues or eliminated. c. Explain why the confidence intervals of Parts (a) and (b) are not the same width even though they both have a confidence level of \(95 \%\).

Example \(9.3\) gave the following airborne times for United Airlines flight 448 from Albuquerque to Denver on 10 randomly selected days: \(\begin{array}{llllllllll}57 & 54 & 55 & 51 & 56 & 48 & 52 & 51 & 59 & 59\end{array}\) a. Compute and interpret a \(90 \%\) confidence interval for the mean airborne time for flight 448 . b. Give an interpretation of the \(90 \%\) confidence level associated with the interval estimate in Part (a). c. Based on your interval in Part (a), if flight 448 is scheduled to depart at 10 A.M., what would you recommend for the published arrival time? Explain.

One thousand randomly selected adult Americans participated in a survey conducted by the Associated Press (June, 2006). When asked "Do you think it is sometimes justified to lie or do you think lying is never justified?" \(52 \%\) responded that lying was never justified. When asked about lying to avoid hurting someone's feelings, 650 responded that this was often or sometimes OK. a. Construct a \(90 \%\) confidence interval for the proportion of adult Americans who think lying is never justified. b. Construct a \(90 \%\) confidence interval for the proportion of adult American who think that it is often or sometimes OK to lie to avoid hurting someone's feelings. c. Based on the confidence intervals from Parts (a) and (b), comment on the apparent inconsistency in the responses given by the individuals in this sample.
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