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Chapter 5: 5-Q38E (page 320)

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.)

Short Answer

Expert verified

A random sample is a subgroup of people chosen at random by investigators to represent the full population as a whole. A technique for selecting a sample of data from a community to make predictions regarding the community.

Step by step solution

01

(a) The data is given below

The calculation is given below:

Given,

n = 80p = 0.3

Meanp= 0.3σP⏜=pqn=0.3×1-0.380=0.0512

≃0.05

02

(b) The data is given below

The calculation is given below:

np = 80×0.3 = 24⩾10nq = 80×0.7 = 56⩾10

Since np⩾10 and nq⩾10,the sample probability is normal distribution is normally distributed with μP⏜=0.3 andσP⏞= 0.05

03

(c) The data is given below

The calculation is given below:

ForP⏞=0.35, the Z score is:

Z=P⏞-μP⏞σP⏞=0.35-0.30.05=1

04

(d) The data is given below

The calculation is given below:

PP⏜>.35=PP⏞-μP⏜σP⏞⩾0.35-μp⏜σp⏜=PZ⩾1=1-PZ<1

=1-0.8413=0.1587

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

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Suppose a random sample of n measurements is selected from a population with u=100mean and variance role="math" localid="1657967387987" σ2=100. For each of the following values of n, give the mean and standard deviation of the sampling distribution of the sample mean.

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