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ra1l [238]
2 years ago
9

83 random samples were selected from a normally distributed population and were found to have a mean of 32.1 and a standard devi

ation of 2.4. Construct a 90% confidence interval for the population standard deviation.
Mathematics
1 answer:
arlik [135]2 years ago
5 0

Answer:

\frac{(82)(2.4)^2}{104.139} \leq \sigma^2 \leq \frac{(82)(2.4)^2}{62.132}

4.525 \leq \sigma^2 \leq 7.602

Now we just take square root on both sides of the interval and we got:

2.127 \leq \sigma \leq 2.757

Step-by-step explanation:

Information given

\bar X=32.1 represent the sample mean

\mu population mean (variable of interest)

s=2.4 represent the sample standard deviation

n=83 represent the sample size  

Confidence interval

The confidence interval for the population variance is given by the following formula:

\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}

The degrees of freedom given by:

df=n-1=8-1=7

The confidence level is 0.90 or 90%, the value of \alpha=0.1 and \alpha/2 =0.05, and we can use excel, a calculator or a table to find the critical values.  

The excel commands would be: "=CHISQ.INV(0.05,82)" "=CHISQ.INV(0.95,82)". so for this case the critical values are:

\chi^2_{\alpha/2}=104.139

\chi^2_{1- \alpha/2}=62.132

The confidence interval is given by:

\frac{(82)(2.4)^2}{104.139} \leq \sigma^2 \leq \frac{(82)(2.4)^2}{62.132}

4.525 \leq \sigma^2 \leq 7.602

Now we just take square root on both sides of the interval and we got:

2.127 \leq \sigma \leq 2.757

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Answer:

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y=1083.48+23.73x

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Step-by-step explanation:

The general form of a linear regression is:

y=a+bx

Here,

<em>y</em> = dependent variable

<em>x</em> = independent variable

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Compute the values of <em>a</em> and <em>b</em> as follows:

\begin{aligned}        a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} =             \frac{ 6857 \cdot 55 - 15 \cdot 17558}{ 6 \cdot 55 - 15^2} \approx 1083.476 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}        = \frac{ 6 \cdot 17558 - 15 \cdot 6857 }{ 6 \cdot 55 - \left( 15 \right)^2} \approx 23.743\end{aligned}

The linear regression equation that represents  the set of data is:

y=1083.48+23.73x

Compute the predicted value of the number of new cases for 2013 (i.e. <em>x</em> = 12) as follows:

y=1083.48+23.73x

  =1083.48+(23.73\times12)\\\\=1083.48+284.76\\\\=1368.24\\\\\approx 1368

Thus, the predicted number of new cases for 2013 is 1368.

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