Mathematics, 23.03.2020 19:32, sherlock19

The diameter of ball bearings are known to be normally distributed with unknown mean and variance. A random sample of size 25 gave a mean of 2.5 cm. The 95% confidence interval had length 4 cm. Then

(a) The sample variance is 4.86.
(b) The sample variance is 26.03.
(c) The population variance is 4.84.
(d) The population variance is 23.47. (e) The sample variance is 23.47.

Answers: 2

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Answers

answered: munekalove69ounxwv

$ME= \frac{4}{2}= 2$

And the margin of error is given by:

$ME= t_{\alpha/2}\frac{s}{\sqrt{n}}$

$s= \frac{ME \sqrt{n}}{t_{\alpha/2}}= \frac{2* \sqrt{25}}{2.0639}= 4.845$

And the sample variance is

$s^2 = 4.845^2 \approx 23.47$

So then the correct option would be:

(e) The sample variance is 23.47.

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 means 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.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X=2.5$ represent the sample mean

$\mu$ population mean (variable of interest)

s represent the sample standard deviation

n=25 represent the sample size

Solution to the problem

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

df=n-1=25-1=24

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,24)".And we see that $t_{\alpha/2}=2.0639$

The lenght of the interval is given by:

2ME = 4

$ME= \frac{4}{2}= 2$

And the margin of error is given by:

$ME= t_{\alpha/2}\frac{s}{\sqrt{n}}$

And solving for the deviation we got:

$s= \frac{ME \sqrt{n}}{t_{\alpha/2}}= \frac{2* \sqrt{25}}{2.0639}= 4.845$

And the sample variance is

$s^2 = 4.845^2 \approx 23.47$

So then the correct option would be:

(e) The sample variance is 23.47.

answered: Guest

answer: i'm not going to give you the answer, but some hints to work it out.

step-by-step explanation: so, if there is a straight line that means that the line angle is 180°. so we subtract the 2 angle measures provided, (15 and 38) from 180. so the answer is 180 - (38 + 15). (you have to solve the equation)

Find the measure of the missing angle pls me plsssss

answered: Guest

answer: 1722

step-by-step explanation:

given: the mass of saturn= $5.685\times10^{26}\ kgs$

the mass of mercury= $3.302\times10^{23}\ kgs$

the number of times the mass of saturn is greater than the mass of mercury

$=\frac{\text{mass of saturn}}{\text{mass of mercury}}\\=\frac{5.685\times10^{26}}{3.302\times10^{23}}\\\\=1.72168\times10^{26-23}{by using law of exponent}]\\=1.72168\times10^3=1721.68\approx1722$