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konstantin123 [22]
3 years ago
14

help me please !!!!!!!!!

Mathematics
1 answer:
azamat3 years ago
3 0

Answer:

A

Step-by-step explanation:

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A random sample from a normal population is obtained, and the data are given below. Find a 90% confidence interval for . 114 157
melamori03 [73]

Answer:

103.160 \leq \sigma \leq 187.476

And the upper bound rounded to the nearest integer would be 187.

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population mean or variance lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.

The Chi Square distribution is the distribution of the sum of squared standard normal deviates .

\bar X represent the sample mean for the sample  

\mu population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

Data given: 114 157 203 257 284 299 305 344 378 410 421 450 478 480 512 533 545

The confidence interval for the population variance \sigma^2 is given by the following formula:

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

On this case we need to find the sample standard deviation with the following formula:

s=sqrt{\frac{\sum_{i=1}^{17} (x_i -\bar x)^2}{n-1}}

And in order to find the sample mean we just need to use this formula:

\bar x =\frac{\sum_{i=1}^n x_i}{n}

The sample mean obtained on this case is \bar x= 362.941 and the deviation s=132.250

The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:

df=n-1=17-1=16

Since the Confidence 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 tabel to find the critical values.  

The excel commands would be: "=CHISQ.INV(0.05,16)" "=CHISQ.INV(0.95,16)". so for this case the critical values are:

\chi^2_{\alpha/2}=26.296

\chi^2_{1- \alpha/2}=7.962

And replacing into the formula for the interval we got:

\frac{(16)(132.250)^2}{26.296} \leq \sigma \leq \frac{(16)(132.250)^2}{7.962}

10641.959 \leq \sigma^2 \leq 35147.074

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

103.160 \leq \sigma \leq 187.476

And the upper bound rounded to the nearest integer would be 187.

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