High School

SAT Scores:

A college admissions officer sampled 116 entering freshmen and found that 45 of them scored more than 590 on the math SAT.

Part 1 of 3

(a) Find a point estimate for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT. Round the answer to at least three decimal places.

The point estimate for the proportion is 0.388.

Part 2 of 3

(b) Construct a 98% confidence interval for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT. Round the answer to at least three decimal places.

Answer :

The 98% confidence interval for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT is approximately 0.283 to 0.493.

To find the point estimate for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT, we divide the number of freshmen who scored more than 590 by the total sample size.

Point Estimate = Number of freshmen who scored more than 590 / Total sample size

In this case, the number of freshmen who scored more than 590 on the math SAT is 45, and the total sample size is 116.

Point Estimate = 45 / 116 ≈ 0.388

Rounded to three decimal places, the point estimate for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT is approximately 0.388.

To construct a 98% confidence interval for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT, we can use the following formula:

Confidence Interval = Point Estimate ± (Critical Value * Standard Error)

The critical value corresponds to the desired confidence level and is obtained from the standard normal distribution. For a 98% confidence level, the critical value is approximately 2.326.

The standard error can be calculated using the following formula:

Standard Error = sqrt((Point Estimate * (1 - Point Estimate)) / Sample Size)

Using the point estimate from part (a) as 0.388 and the sample size as 116, we can calculate the standard error:

Standard Error = sqrt((0.388 * (1 - 0.388)) / 116) ≈ 0.050

Now we can construct the confidence interval:

Confidence Interval = 0.388 ± (2.326 * 0.050)

Lower Bound = 0.388 - (2.326 * 0.050) ≈ 0.283

Upper Bound = 0.388 + (2.326 * 0.050) ≈ 0.493

Rounded to three decimal places, the 98% confidence interval for the proportion of all entering freshmen at this college who scored more than 590 on the math SAT is approximately 0.283 to 0.493.

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