Question

Suppose x has a distribution with a mean of 70 and a standard deviation of 44. Random samples of size 64 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the x distribution. x has a distribution with mean μx and standard deviation σx. (b) Find the z-value corresponding to x = 75.5. (c) Find P(x > 75.5). (Round your answer to four decimal places.) (d) Would it be unusual for a random sample of size 64 from the x distribution to have a sample mean less than 75.5? Explain. It would be unusual because more than 5% of all such samples have means less than 75.5. No, it would not be unusual because less than 5% of all such samples have means less than 75.5. No, it would not be unusual because more than 5% of all such samples have means less than 75.5. Yes, it would be unusual because less than 5% of all such samples have means less than 75.5.

          Suppose x has a distribution with a mean of 70 and a standard deviation of 44. Random samples of size 64 are drawn.

(a) Describe the x distribution and compute the mean and standard deviation of the x distribution. x has a distribution with mean μx and standard deviation σx.
(b) Find the z-value corresponding to x = 75.5.
(c) Find P(x > 75.5). (Round your answer to four decimal places.)
(d) Would it be unusual for a random sample of size 64 from the x distribution to have a sample mean less than 75.5? Explain.
It would be unusual because more than 5% of all such samples have means less than 75.5.
No, it would not be unusual because less than 5% of all such samples have means less than 75.5.
No, it would not be unusual because more than 5% of all such samples have means less than 75.5.
Yes, it would be unusual because less than 5% of all such samples have means less than 75.5.

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Added by John S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Joanna Quigley

08/10/2023

Step 1

We are given a sample size of 64. (a) The distribution of the sample means will be approximately normal due to the Central Limit Theorem. The mean of the sample means will be equal to the population mean, and the standard deviation of the sample means will be the Show more…

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Suppose x has a distribution with a mean of 70 and a standard deviation of 44. Random samples of size 64 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the x distribution. x has a distribution with mean μx and standard deviation σx. (b) Find the z-value corresponding to x = 75.5. (c) Find P(x > 75.5). (Round your answer to four decimal places.) (d) Would it be unusual for a random sample of size 64 from the x distribution to have a sample mean less than 75.5? Explain. It would be unusual because more than 5% of all such samples have means less than 75.5. No, it would not be unusual because less than 5% of all such samples have means less than 75.5. No, it would not be unusual because more than 5% of all such samples have means less than 75.5. Yes, it would be unusual because less than 5% of all such samples have means less than 75.5.

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Transcript

00:01 We are starting with a population that has a mean mu of 70 and a standard deviation sigma of 44.

00:07 But we aren't looking at individual observations, we are looking at a sample of size 64.

00:13 So part a we want to describe this distribution.

00:17 Now i don't know the shape of the original population distribution, but i do know if i take every possible sample of size 64, take the means and plot them out, i get something approximately normal.

00:30 This is because of the central limit theorem, which states that as sample size increases, sample means become more and more normally distributed.

00:39 If n is at least 30, you can treat them as approximately normal.

00:43 The mean of the means is the same as the population mean.

00:47 The standard deviation of the sample means, or standard error, is sigma over root n.

00:53 So if i want to describe this distribution, it is approximately normal.

01:00 It has a mean of 70, a standard deviation 44 over root 64 is 5 .5.

01:10 Part b, what value corresponds to a sample mean of 75 .5? so we're looking for the corresponding z -score.

01:20 This tells you how many standard deviations away from the mean a value is.

01:24 It's the standardized score.

01:26 You get it by taking the value, here a sample mean, subtracting the mean of its distribution, dividing by the standard deviation of its distribution.

01:34 Here that is 1.

01:38 It is one standard error above the mean.

01:45 So what we're saying is that 75 .5 on this sampling distribution corresponds to a value of 1 on the standard normal curve.

01:55 Now we want the probability that the sample mean would be less than this value.

02:01 Okay, so to find this you need something with the normal distribution built in.

02:06 The functions have to integrate.

02:07 To do this by hand is just too complicated...

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