Solved on Jan 02, 2024

A college admissions officer takes a random sample of 100 entering freshmen and finds their mean mathematics SAT score is 436. Assuming the population standard deviation is $\sigma=115$ , construct a $99\%$ confidence interval for the mean mathematics SAT score of the entering freshman class, rounded to the nearest whole number.
A $99\%$ confidence interval for the mean mathematics SAT score is $\boxed{395}<\mu<\boxed{477}$ .

STEP 1

Assumptions
1. The sample size is $n = 100$ .
2. The sample mean is $\bar{x} = 436$ .
3. The population standard deviation is $\sigma = 115$ .
4. We are constructing a $99\%$ confidence interval for the population mean $\mu$ .
5. Since the sample size is large ( $n \geq 30$ ), we can use the z-distribution to approximate the sampling distribution of the sample mean.

STEP 2

Determine the z-score that corresponds to a $99\%$ confidence level. This value is the critical value $z_{\alpha/2}$ , where $\alpha = 1 - 0.99 = 0.01$ and $\alpha/2 = 0.005$ .

STEP 3

Look up the critical z-score in a standard normal (z) distribution table or use a statistical calculator to find $z_{0.005}$ .

STEP 4

The z-score that corresponds to a $99\%$ confidence level (with $\alpha/2 = 0.005$ ) is approximately $z_{\alpha/2} = 2.576$ .

STEP 5

Calculate the margin of error using the formula:
$Margin\, of\, Error = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

STEP 6

Plug in the values for $z_{\alpha/2}$ , $\sigma$ , and $n$ to calculate the margin of error.
$Margin\, of\, Error = 2.576 \times \frac{115}{\sqrt{100}}$

STEP 7

Since $\sqrt{100} = 10$ , simplify the margin of error calculation.
$Margin\, of\, Error = 2.576 \times \frac{115}{10}$

STEP 8

Calculate the margin of error.
$Margin\, of\, Error = 2.576 \times 11.5$

STEP 9

Calculate the margin of error.
$Margin\, of\, Error = 29.624$

STEP 10

Round the margin of error to the nearest whole number.
$Margin\, of\, Error \approx 30$

STEP 11

Construct the confidence interval using the sample mean and the margin of error.
$Lower\, limit = \bar{x} - Margin\, of\, Error$ $Upper\, limit = \bar{x} + Margin\, of\, Error$

STEP 12

Calculate the lower limit of the confidence interval.
$Lower\, limit = 436 - 30$

STEP 13

Calculate the lower limit of the confidence interval.
$Lower\, limit = 406$

STEP 14

Calculate the upper limit of the confidence interval.
$Upper\, limit = 436 + 30$

STEP 15

Calculate the upper limit of the confidence interval.
$Upper\, limit = 466$
The $99\%$ confidence interval for the mean mathematics SAT score for the entering freshman class is $406 < \mu < 466$ .

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STEP 1

STEP 2

Determine the z-score that corresponds to a $99\%$ confidence level. This value is the critical value $z_{\alpha/2}$ , where $\alpha = 1 - 0.99 = 0.01$ and $\alpha/2 = 0.005$ .

STEP 3

Look up the critical z-score in a standard normal (z) distribution table or use a statistical calculator to find $z_{0.005}$ .

STEP 4

The z-score that corresponds to a $99\%$ confidence level (with $\alpha/2 = 0.005$ ) is approximately $z_{\alpha/2} = 2.576$ .

STEP 5

Calculate the margin of error using the formula:
$Margin\, of\, Error = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

STEP 6

Plug in the values for $z_{\alpha/2}$ , $\sigma$ , and $n$ to calculate the margin of error.
$Margin\, of\, Error = 2.576 \times \frac{115}{\sqrt{100}}$

STEP 7

Since $\sqrt{100} = 10$ , simplify the margin of error calculation.
$Margin\, of\, Error = 2.576 \times \frac{115}{10}$

STEP 8

Calculate the margin of error.
$Margin\, of\, Error = 2.576 \times 11.5$

STEP 9

Calculate the margin of error.
$Margin\, of\, Error = 29.624$

STEP 10

Round the margin of error to the nearest whole number.
$Margin\, of\, Error \approx 30$

STEP 11

Construct the confidence interval using the sample mean and the margin of error.
$Lower\, limit = \bar{x} - Margin\, of\, Error$ $Upper\, limit = \bar{x} + Margin\, of\, Error$

STEP 12

Calculate the lower limit of the confidence interval.
$Lower\, limit = 436 - 30$

STEP 13

Calculate the lower limit of the confidence interval.
$Lower\, limit = 406$

STEP 14

Calculate the upper limit of the confidence interval.
$Upper\, limit = 436 + 30$

STEP 15

Calculate the upper limit of the confidence interval.
$Upper\, limit = 466$
The $99\%$ confidence interval for the mean mathematics SAT score for the entering freshman class is $406 < \mu < 466$ .

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STEP 1

STEP 2

Determine the z-score that corresponds to a 99%99\%99% confidence level. This value is the critical value zα/2z_{\alpha/2}zα/2​, where α=1−0.99=0.01\alpha = 1 - 0.99 = 0.01α=1−0.99=0.01 and α/2=0.005\alpha/2 = 0.005α/2=0.005.

STEP 3

Look up the critical z-score in a standard normal (z) distribution table or use a statistical calculator to find z0.005z_{0.005}z0.005​.

STEP 4

The z-score that corresponds to a 99%99\%99% confidence level (with α/2=0.005\alpha/2 = 0.005α/2=0.005) is approximately zα/2=2.576z_{\alpha/2} = 2.576zα/2​=2.576.

STEP 5

Calculate the margin of error using the formula:Margin of Error=zα/2×σnMargin\, of\, Error = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}MarginofError=zα/2​×n​σ​

STEP 6

Plug in the values for zα/2z_{\alpha/2}zα/2​, σ\sigmaσ, and nnn to calculate the margin of error.Margin of Error=2.576×115100Margin\, of\, Error = 2.576 \times \frac{115}{\sqrt{100}}MarginofError=2.576×100​115​

STEP 7

Since 100=10\sqrt{100} = 10100​=10, simplify the margin of error calculation.Margin of Error=2.576×11510Margin\, of\, Error = 2.576 \times \frac{115}{10}MarginofError=2.576×10115​

STEP 8

Calculate the margin of error.Margin of Error=2.576×11.5Margin\, of\, Error = 2.576 \times 11.5MarginofError=2.576×11.5

STEP 9

Calculate the margin of error.Margin of Error=29.624Margin\, of\, Error = 29.624MarginofError=29.624

STEP 10

Round the margin of error to the nearest whole number.Margin of Error≈30Margin\, of\, Error \approx 30MarginofError≈30

STEP 11

Construct the confidence interval using the sample mean and the margin of error.Lower limit=xˉ−Margin of ErrorLower\, limit = \bar{x} - Margin\, of\, ErrorLowerlimit=xˉ−MarginofError Upper limit=xˉ+Margin of ErrorUpper\, limit = \bar{x} + Margin\, of\, ErrorUpperlimit=xˉ+MarginofError

STEP 12

Calculate the lower limit of the confidence interval.Lower limit=436−30Lower\, limit = 436 - 30Lowerlimit=436−30

STEP 13

Calculate the lower limit of the confidence interval.Lower limit=406Lower\, limit = 406Lowerlimit=406

STEP 14

Calculate the upper limit of the confidence interval.Upper limit=436+30Upper\, limit = 436 + 30Upperlimit=436+30

STEP 15

Calculate the upper limit of the confidence interval.Upper limit=466Upper\, limit = 466Upperlimit=466The 99%99\%99% confidence interval for the mean mathematics SAT score for the entering freshman class is 406<μ<466406 < \mu < 466406<μ<466.

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Determine the z-score that corresponds to a $99\%$ confidence level. This value is the critical value $z_{\alpha/2}$ , where $\alpha = 1 - 0.99 = 0.01$ and $\alpha/2 = 0.005$ .

Look up the critical z-score in a standard normal (z) distribution table or use a statistical calculator to find $z_{0.005}$ .

The z-score that corresponds to a $99\%$ confidence level (with $\alpha/2 = 0.005$ ) is approximately $z_{\alpha/2} = 2.576$ .

Calculate the margin of error using the formula:
$Margin\, of\, Error = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

Plug in the values for $z_{\alpha/2}$ , $\sigma$ , and $n$ to calculate the margin of error.
$Margin\, of\, Error = 2.576 \times \frac{115}{\sqrt{100}}$

Since $\sqrt{100} = 10$ , simplify the margin of error calculation.
$Margin\, of\, Error = 2.576 \times \frac{115}{10}$

Calculate the margin of error.
$Margin\, of\, Error = 2.576 \times 11.5$

Calculate the margin of error.
$Margin\, of\, Error = 29.624$

Round the margin of error to the nearest whole number.
$Margin\, of\, Error \approx 30$

Construct the confidence interval using the sample mean and the margin of error.
$Lower\, limit = \bar{x} - Margin\, of\, Error$ $Upper\, limit = \bar{x} + Margin\, of\, Error$

Calculate the lower limit of the confidence interval.
$Lower\, limit = 436 - 30$

Calculate the lower limit of the confidence interval.
$Lower\, limit = 406$

Calculate the upper limit of the confidence interval.
$Upper\, limit = 436 + 30$

Calculate the upper limit of the confidence interval.
$Upper\, limit = 466$
The $99\%$ confidence interval for the mean mathematics SAT score for the entering freshman class is $406 < \mu < 466$ .