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Chapter 5: Q36E (page 320)

Suppose a random sample of n measurements is selected from a binomial population with the probability of success p = .2. For each of the following values of n, give the mean and standard deviation of the sampling distribution of the sample proportion,

  1. n = 50
  2. n = 1,000
  3. n = 400

Short Answer

Expert verified

A sampling distribution is a statistic that calculates the chance of an occurrence depending on information from a tiny subset of a significant population.

Step by step solution

01

(a) The n = 50 calculations are given below

For various values of n, we must calculate the mean as well as standard deviations of the sampling range of the probability value. If we consider p as a proportion, the sample mean may be regarded as a normal distribution.

The calculation is given below:

Mean=pStandardDeviation=PQnP=Numberofsuccess.Q=1-P=Numberoffailures.

localid="1662358414393" n=50Mean=0.2StandardDeviation=PQn=0.2×0.850

=0.056

02

(b) The n = 1,000 calculations are given below

The calculation is given below:

n=1000Mean=0.2StandardDeviation=PQn=0.2×0.81000

=0.0126

03

(c) The n = 400 calculations are given below

The calculation is given below:

n=400Mean=0.2StandardDeviation=PQn=0.2×0.8400

=0.02

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Most popular questions from this chapter

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 (χ¯) and median m for each sample. Construct relative frequency histograms for the 500 values of (χ¯)and the 500 values of m. Use these approximations to the sampling distributions of (χ¯)and m to answer the following questions:

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

b. Which sampling distribution displays greater variation?

Analysis of supplier lead time. Lead timeis the time betweena retailer placing an order and having the productavailable to satisfy customer demand. It includes time for placing the order, receiving the shipment from the supplier, inspecting the units received, and placing them in inventory. Interested in average lead time,, for a particular supplier of men’s apparel, the purchasing department of a national department store chain randomly sampled 50 of the supplier’s lead times and found= 44 days.

  1. Describe the shape of the sampling distribution ofx¯.
  2. If μand σare really 40 and 12, respectively, what is the probability that a second random sample of size 50 would yieldx¯ greater than or equal to 44?
  3. Using the values forμ and σin part b, what is the probability that a sample of size 50 would yield a sample mean within the interval μ±2σn?

Will the sampling distribution of x¯always be approximately normally distributed? Explain

Plastic fill process. University of Louisville operators examined the process of filling plastic pouches of dry blended biscuit mix (Quality Engineering, Vol. 91, 1996). The current fill mean of the process is set at μ= 406 grams, and the process fills standard deviation is σ= 10.1 grams. (According to the operators, “The high level of variation is since the product has poor flow properties and is, therefore, difficult to fill consistently from pouch to pouch.”) Operators monitor the process by randomly sampling 36 pouches each day and measuring the amount of biscuit mix in each. Considerx the mean fill amount of the sample of 36 products. Suppose that on one particular day, the operators observe x= 400.8. One of the operators believes that this indicates that the true process fill mean for that day is less than 406 grams. Another operator argues thatμ = 406, and the small observed value is due to random variation in the fill process. Which operator do you agree with? Why?

A random sample of n = 80 measurements is drawn from a binomial population with a probability of success .3.

  1. Give the mean and standard deviation of the sampling distribution of the sample proportion,P^
  2. Describe the shape of the sampling distribution ofP⏜
  3. Calculate the standard normal z-score corresponding to a value ofP⏜=.35.
  4. FindP(P⏜=.35.)
See all solutions

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