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In a random sample of 80 teenagers, the average number of texts handled in a day is 50. The 96% confidence interval for the mean

Nastasia [14]

Answer:

a) $\bar X =\frac{46+54}{2}=50$

And the margin of error is given by:

$ME= \frac{54-46}{2}= 4$

The confidence level is 0.96 and the significance level is $\alpha=1-0.96=0.04$ and the value of $\alpha/2 =0.02$ and the margin of error is given by:

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

We can calculate the critical value and we got:

$z_{\alpha/2} = 2.05$

And if we solve for the deviation like this:

$\sigma = ME * \frac{\sqrt{n}}{z_{\alpha/2}}$

And replacing we got:

$\sigma =4 *\frac{\sqrt{80}}{2.05} =17.45$

b) $ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=2.05 *\frac{17.45}{\sqrt{160}}=2.828$

And as we can see that the margin of error would be lower than the original value of 4, the margin of error would be reduced by a factor $\sqrt{2}$

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".

$\bar X$ represent the sample mean

$\mu$ population mean (variable of interest)

$\sigma$ represent the population standard deviation

n=80 represent the sample size

Solution to the problem

Part a

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

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

For this case we can calculate the mean like this:

$\bar X =\frac{46+54}{2}=50$

And the margin of error is given by:

$ME= \frac{54-46}{2}= 4$

The confidence level is 0.96 and the significance level is $\alpha=1-0.96=0.04$ and the value of $\alpha/2 =0.02$ and the margin of error is given by:

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

We can calculate the critical value and we got:

$z_{\alpha/2} = 2.05$

And if we solve for the deviation like this:

$\sigma = ME * \frac{\sqrt{n}}{z_{\alpha/2}}$

And replacing we got:

$\sigma =4 *\frac{\sqrt{80}}{2.05} =17.45$

Part b

For this case is the sample size is doubled the margin of error would be:

$ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=2.05 *\frac{17.45}{\sqrt{160}}=2.828$

And as we can see that the margin of error would be lower than the original value of 4, the margin of error would be reduced by a factor $\sqrt{2}$

5 0

2 years ago

If a rectangular poster has an area of

d1i1m1o1n [39]

$2x ^2 - 9x + 10=\\
2x^2-4x-5x+10=\\
2x(x-2)-5(x-2)=\\
(2x-5)(x-2) \Rightarrow C$

8 0

2 years ago

the distant around a track is 1/4 mile. a marathon is 26 1/5 miles. How many laps does it take to run a marathon

sleet_krkn [62]

Answer:

108 1/8

Step-by-step explanation:

1/4=0.25, 1/5=0.2

=>26+0.2=26.2

=>26.2/0.25=108.8

8 0

3 years ago

Identify whether the series summation of 12 open parentheses 3 over 5 close parentheses to the I minus 1 power from 1 to infinit

Finger [1]

∑ from 1 to infinity of 12(3/5)^(i - 1)
Since the common ratio is less than 1, the series is convegent. [i.e. 3/5 < 1]

Sum to infinity of a geometric series is given by a/(1 - r); where a is the first term, and r is the common ratio.

Sum = 12/(1 - 3/5) = 12/(2/5) = 30.

3 0

2 years ago

Find the unknown side length of the rectangle if its area is 11 square feet.

expeople1 [14]

Answer:

3

Step-by-step explanation:

- Area of rectangle = length * width

$11 = 3\frac{2}{3} *w\\\\- rearrange and divide\\ 11 \\ by \\\\ 3\frac{2}{3}\\\\ = 3$

6 0

2 years ago

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