Question

answer all 3 for best answer

1.The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth-graders. Scores on the test range from 0 to 500. Suppose that you give the NAEP test to an SRS of 1225 eighth-graders from a large population in which the scores have mean μ = 281 and standard deviation σ = 129. The mean x will vary if you take repeated samples.

The sampling distribution of x is approximately Normal. It has mean μ = 281. What is its standard deviation? (Round your answer to three decimal places.)

2.)

A recent survey describes the total sleep time per night among college students as approximately Normally distributed with mean μ = 6.76 hours and standard deviation σ = 1.26 hours. You initially plan to take an SRS of size

n = 185

and compute the average total sleep time.

(a)

What is the standard deviation for the average time in hours? (Round your answer to four decimal places.)

hr

in minutes? (Round your answer to three decimal places.)

min

(b)

Use the 95 part of the 68–95–99.7 rule to describe the variability of this sample mean. (Round your answer to four decimal places.)

95% of the time the sample mean will be between a lower value of  hours and an upper value of  hours.

(c)

What is the probability that your average will be below 6.9 hours? (Round your answer to four decimal places.)

3.)Suppose that scores on the mathematics part of a test for eighth-grade students follow a Normal distribution with standard deviation σ = 150. You want to estimate the mean score within ±12 with 90% confidence. How large an SRS of scores must you choose? (Round your answer up to the next whole number.)
scores

You may need to use the appropriate Appendix Table to answer this question.

Answer 1

1]

The sampling distribution of x is approximately Normal. It has mean μ = 281.

And its standard deviation σ = 129.000

2]

The average total sleep time = 6.76 hours

a]

standard deviation for the average time in hours : = 0.0926 hours

standard deviation for the average time in minures : minutes.

b]

95% of the time the sample mean will be between a lower value of  hours and an upper value of  hours:

where

Lower value: = 6.76 - 1.96 * 0.0926 = 6.5784

Upper value : = 6.76 + 1.96 * 0.0926 = 6.9416

c]

the probability that your average will be below 6.9 hours = Pr( X < 6.9) = 0.5442

Using Excel command " =NORM.DIST(6.9,6.76,1.26,TRUE) " , we get the probability that your average will be below 6.9 hours = 0.5442.

3]

We have lower value and upper value of population mean, -12 and +12 respectively.

We know that, = Upper value + Lower value = 12 -12 = 0

==> = 0

That is estimated the mean score = 0.

Now

How large an SRS of scores must you choose, we have to find sample size n

We know that the formula of Upper value: = 12

= 12    Where

= 20.5607

Squaring on both sides , we get

n = 422.7412 but nearest whole number is 423.

So, n = 423.