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LekaFEV [45]
3 years ago
15

In a large lecture class, the professor announced that the scores on a recent exam were normally distributed with a range from 5

1 to 87. Using the Empirical Rule μ ± 3σ to estimate σ, how many students would you need to sample to estimate the true mean score for the class with 90 percent confidence and an error of ±2?
Mathematics
1 answer:
Mars2501 [29]3 years ago
4 0

Answer:

The standard deviation of the scores on a recent exam is 6.

The sample size required is 25.

Step-by-step explanation:

Let <em>X</em> = scores of students on a recent exam.

It is provided that the random variable <em>X</em> is normally distributed.

According to the Empirical rule, 99.7% of the normal distribution is contained in the range, <em>μ </em>± 3<em>σ</em>.

That is, P (<em>μ </em>- 3<em>σ </em>< <em>X</em> < <em>μ </em>+ 3<em>σ</em>) = 0.997.

It is provided that the scores on a recent exam were normally distributed with a range from 51 to 87.

This implies that:

P (51 < <em>X</em> < 87) = 0.997

So,

<em>μ </em>- 3<em>σ </em>= 51...(i)

<em>μ </em>+ 3<em>σ </em>= 87...(ii)

Subtract (i) and (ii) to compute the value of <em>σ</em> as follows:

<em>   μ </em>-     3<em>σ </em>=    51

(-)<em>μ </em>+ (-)3<em>σ </em>= (-)87

______________

-6<em>σ </em>= -36

<em>σ</em> = 6

Thus, the standard deviation of the scores on a recent exam is 6.

The (1 - <em>α</em>)% confidence interval for population mean is given by:

CI=\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}

The margin of error of this interval is:

MOE = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}

Given:

MOE = 2

<em>σ</em> = 6

Confidence level = 90%

Compute the <em>z</em>-score for 90% confidence level as follows:

z_{\alpha/2}=z_{0.10/2}=z_{0.05}=1.645

*Use a <em>z</em>-table.

Compute the sample required as follows:

MOE = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\\2=1.645\times \frac{6}{\sqrt{n}}\\n=(\frac{1.645\times 6}{2})^{2}\\n=24.354225\\n\approx 25

Thus, the sample size required is 25.

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