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Chapter 5: Q37E (page 320)

Suppose a random sample of n = 500 measurements is selected from a binomial population with probability of success p. For each of the following values of p, give the mean and standard deviation of the sampling distribution of the sample proportion,p^.

  1. p= .1
  2. p= .5
  3. p= .7

Short Answer

Expert verified

The mean and standard deviation of the sampling distribution of the sampling proportion are,

  1. If p=0.1, mean = 0.1, standard deviation = 0.0134
  2. If p=0.5, mean = 0.5, standard deviation = 0.0223
  3. If p=0.7, mean = 0.7, standard deviation = 0.0204

Step by step solution

01

Given information

There is a random sample of n=500 from a binomial distribution with a probability of success p.

02

Calculate the mean and standard deviation when p=.1

The mean of the sampling distribution is equal to the true binomial proportion. So, and the standard deviation of the sampling distribution is σp^=p(1-p)/n.

Therefore,

a.

If p=0.1 then, Mean Ep^=p=0.1.

Standard deviationσp^=0.11-0.1/500=0.0134

03

Calculate the mean and standard deviation p=.5

b.

If p=0.5 then, MeanEp^=p=0.5

Standard deviationσp^=0.51-0.5/500=0.0223

04

Calculate the mean and standard deviation p=.7

c.

If,p=0.7 then, MeanEp^=p=0.7

Standard deviationσp^=0.71-0.7/500=0.0204

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Most popular questions from this chapter

Critical-part failures in NASCAR vehicles. Refer to The Sport Journal (Winter 2007) analysis of critical-part failures at NASCAR races, Exercise 4.144 (p. 277). Recall that researchers found that the time x (in hours) until the first critical-part failure is exponentially distributed with μ= .10 and s = .10. Now consider a random sample of n = 50 NASCAR races and let χ¯ represent the sample meantime until the first critical-part failure.

a) Find E(χ¯) and Var(χ¯)

b) Although x has an exponential distribution, the sampling distribution of x is approximately normal. Why?

c) Find the probability that the sample meantime until the first critical-part failure exceeds .13 hour.

Refer to Exercise 5.18. Find the probability that

  1. x¯is less than 16.
  2. x¯is greater than 23.
  3. x¯is greater than 25.
  4. x¯falls between 16 and 22.
  5. x¯ is less than 14.

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 of p¯
  3. Calculate the standard normal z-score corresponding to a value of p¯=0.35
  4. FindP(p)¯=0.35

:A random sample of n = 68 observations is selected from a population withμ=19.6and σ=3.2Approximate each of the following probabilities

a)pX¯⩽19.6

b)pX¯⩽19

c)pX¯⩾20.1

d)p19.2⩽X¯⩽20.6

Suppose a random sample of n = 25 measurements are selected from a population with mean μand standard deviation s. For each of the following values of μand role="math" localid="1651468116840" σ, give the values of μχ¯ and σχ¯.

  1. μ=100,σ=3
  2. μ=100,σ=25
  3. μ=20,σ=40
  4. μ=10,σ=100
See all solutions

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