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Chapter 5: Q18E (page 314)

A random sample of n = 64 observations is drawn from a population with a mean equal to 20 and a standard deviation equal to 16

a. Give the mean and standard deviation of the (repeated) sampling distribution of x.

b. Describe the shape of the sampling distribution of x. Does your answer depend on the sample size?

c. Calculate the standard normal z-score corresponding to a value of x = 15.5.

d. Calculate the standard normal z-score corresponding to x = 23

Short Answer

Expert verified

Answer

The standard deviation is commonly utilized as a measurement of an asset's comparative volatility. The standard deviation is determined as the square root of the variation by calculating the departure of every observation point from the mean.

Step by step solution

01

Step-by-Step Solution Step 1: (a) The data is given below

The calculation is given below:

渭X=20蟽X=蟽n=1664=2

02

(b) The data is given below

Step 3: (c) The data is given below of x will have the form of a bell-shaped normally distributed. Yes, it is dependent on the scale of the sample. As the sample size improves the sample size approaches being normal.

03

(c) The data is given below

The calculation is given below:

z =X鈭�渭蟽/n=15.5鈭�202=鈭�2.25

04

(d) The data is given below

The calculation is given below:

z =X鈭�渭蟽/n=23鈭�202=1.5

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Most popular questions from this chapter

Levelness of concrete slabs. Geotechnical engineers use water-level "manometer" surveys to assess the levelness of newly constructed concrete slabs. Elevations are typically measured at eight points on the slab; the maximum differential between elevations is of interest. The Journal of Performance of Constructed Facilities (February 2005) published an article on the levelness of slabs in California residential developments. Elevation data collected for more than 1,300 concrete slabs before tensioning revealed that the maximum differential, x, has a mean of 渭=0.53an inch and a standard deviation of 蟽=0.193an inch. Consider a sample of n = 50 slabs selected from those surveyed, and letX represent the sample's mean.

  1. Fully describe the sample sampling distribution of x.
  2. FindP(x炉>0.58)
  3. The study also revealed that the mean maximum differential of concrete slabs measured after tensioning and loading is渭=0.58 an inch. Suppose the sample data yieldx炉=0.59 is an inch. Comment on whether the sample measurements were obtained before or after tensioning and loading.

Question:Quality control. Refer to Exercise 5.68. The mean diameter of the bearings produced by the machine is supposed to be .5 inch. The company decides to use the sample mean from Exercise 5.68 to decide whether the process is in control (i.e., whether it is producing bearings with a mean diameter of .5 inch). The machine will be considered out of control if the mean of the sample of n = 25 diameters is less than .4994 inch or larger than .5006 inch. If the true mean diameter of the bearings produced by the machine is .501 inch, what is the approximate probability that the test will imply that the process is out of control?

Use the computer to generate 500 samples, each containing n = 25 measurements, from a population that contains values of x equal to 1, 2, . . 48, 49, 50 Assume that these values of x are equally likely. Calculate the sample mean (蠂炉) and median m for each sample. Construct relative frequency histograms for the 500 values of (蠂炉)and the 500 values of m. Use these approximations to the sampling distributions of (蠂炉)and m to answer the following questions:

a. Does it appear that and m are unbiased estimators of the population mean? [Note:渭=25.5]

b. Which sampling distribution displays greater variation?

Question:A random sample of 40 observations is to be drawn from a large population of measurements. It is known that 30% of the measurements in the population are 1s, 20% are 2s, 20% are 3s, and 30% are 4s.

a. Give the mean and standard deviation of the (repeated) sampling distribution ofx炉, the sample mean of the 40 observations.

b. Describe the shape of the sampling distribution ofx炉. Does youranswer depend on the sample size?

Refer to Exercise 5.18. Find the probability that

  1. x炉is less than 16.
  2. x炉is greater than 23.
  3. x炉is greater than 25.
  4. x炉falls between 16 and 22.
  5. x炉is less than 14.
See all solutions

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