Question

sat scores: a college admissions officer takes a sample random sample of 100 entering freshmen and computes their mean mathematics sat score to be 482. assume the population standard deviation is σ = 114.
construct a 99.5% confidence interval for the mean mathematics sat score for the entering freshman class. round the answer to the nearest whole number.
a 99.5% confidence interval for the mean mathematics sat score is 478 < μ < 486.

Explanation:

Step1: Calculate the margin of error

For a \(99.5\%\) confidence interval, the z - score \(z\) (from the standard normal distribution table) is \(z = 2.807\). The formula for the margin of error \(E\) when the population standard deviation \(\sigma\) is known is \(E=z\times\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 11.4\), \(n = 100\). Then \(E=2.807\times\frac{11.4}{\sqrt{100}}=2.807\times\frac{11.4}{10}=2.807\times1.14 = 3.2\) (rounded to one decimal place).

Step2: Calculate the sample mean

Let the sample mean be \(\bar{x}\). We know that the confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\). Assume the mid - point of the incorrect interval (if we consider the wrong values \(478\) and \(486\)) was wrong. The mid - point of a confidence interval is \(\bar{x}=\frac{(\text{lower limit}+\text{upper limit})}{2}\). Let's first find the correct mid - point. Since \(n = 100\), \(\sigma=11.4\), \(z = 2.807\). The correct confidence interval formula is \(\bar{x}-z\frac{\sigma}{\sqrt{n}}<\mu<\bar{x}+z\frac{\sigma}{\sqrt{n}}\). Let's assume the sample mean \(\bar{x}\) is calculated as \(\bar{x}=\frac{478 + 486}{2}=482\). Then the correct lower limit is \(\bar{x}-E=482-3.2 = 478.8\) and the correct upper limit is \(\bar{x}+E=482 + 3.2=485.2\)

Answer:

\(478.8<\mu<485.2\)