Section 7.3

Sampling Distribution for Proportions

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Focus Points

• Compute the mean and standard deviation for the sample proportion

• Use the normal approximation to compute probabilities for proportions

• Construct p-Charts and interpret their meaning.

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Sampling Distributions for Proportions

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Sampling Distributions for Proportions

We have studied the sampling distribution for the mean. Now we have the tools to look at sampling distributions for proportions.

Suppose we repeat a binomial experiment with n trials again and again and, for each n trials, record the sample proportion of successes

If np > 5 and nq > 5, the r distribution is approximately normal.

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Sampling Distributions for Proportions

The values form a sampling distribution for proportions.

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Continuity Correction

Procedure: HOW TO MAKE CONTINUITY CORRECTIONS TO INTERVALS

1. If r/n is the right endpoint of a interval, we add 0.5/n to get the corresponding right endpoint of the x interval.

2. 2. If r/n is the left endpoint of a interval, we subtract 0.5/n to get the corresponding left endpoint of the x interval.

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Example 4: Continuity Corrections

Suppose n = 25 and we have a interval from 10/25 = 0.40 to 15/25 = 0.60. Use the continuity correction to convert this interval to an x interval.

Solution:

x-interval: 0.40 − 0.5/25 to 0.60 + 0.5/25 = 0.40 − 0.02 to 0.60 + 0.02 =[0.38, 0.62]

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Example 5 – Sampling distribution of p

The annual crime rate in the Capital Hill neighborhood of Denver is 111 victims per 1000 residents. This means that 111 out of 1000 residents have been the victim of at least one crime.

These crimes range from relatively minor crimes (stolen hubcaps or purse snatching) to major crimes (murder).

^

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Example 5 – Sampling distribution of p

The Arms is an apartment building in this neighborhood that has 50 year round residents. Suppose we view each of the n = 50 residents as a binomial trial.

The random variable r (which takes on values 0, 1, 2, . . . , 50) represents the number of victims of at least one crime in the next year.

^cont’d

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Example 5(a) – Sampling distribution of p

What is the population probability p that a resident in the Capital Hill neighborhood will be the victim of a crime next year? What is the probability q that a resident will not be a victim?

Solution:

Using the Piton Foundation report, we take

p = 111/1000 = 0.111 and q = 1 – p = 0.889

cont’d

^

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Example 5(b) – Sampling distribution of p

Consider the random variable

Can we approximate the distribution with a normal distribution? Explain.

Solution:np = 50(0.111) = 5.55

nq = 50(0.889) = 44.45

Since both np and nq are greater than 5, we can approximate the distribution with a normal distribution.

cont’d

^

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Example 5(c) – Sampling distribution of p

What are the mean and standard deviation for the distribution?

Solution:

cont’d

^

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Example 5(d)

(d) What is the probability that between 10% and 20% of the Arms residents will be victims of a crime next year? Interpret the results.

Solution:

Continuity correction = 0.5/n =0.5/50 = 0.001

P(0.10 ≤ ≤ 0.20) ≈ P(0.09 ≤ x ≤ 0.21) ≈ P(-0.48 ≤ z ≤ 2.25)

≈ 0.6722

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Control Chart for Proportion: P-Chart

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Control Chart for Proportion: P-Chart

Control chart for proportions r/n. Such a chart is often called a P-Chart.

The control charts discussed in Section 6.1 were for

quantitative data, where the size of something is being

measured. There are occasions where we prefer to

examine a quality or attribute rather than just size. One way

to do this is to use a binomial distribution in which success

is defined as the quality or attribute we wish to study.

The basic idea for using P-Charts is to select samples of a

fixed size n at regular time intervals and count the number

of successes r from the n trials.

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We use the normal approximation for r/n and methods of

Section 6.1 to plot control limits and r/n values, and to

interpret results.

As in Section 6.1, we remind ourselves that control charts

are used as warning devices tailored by a user for a

particular need. Our assumptions and probability

calculations need not be absolutely precise to achieve our

purpose. For example, = r/n need not follow a normal

distribution exactly. A mound-shaped and more or less

symmetric distribution to which the empirical rule applies

will be sufficient.

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Example 6: P-Chart

Anatomy and Physiology is taught each semester. The

course is required for several popular health-science

majors, so it always fills up to its maximum of 60 students.

The dean of the college asked the biology department to

make a control chart for the proportion of A’s given in the

course each semester for the past 14 semesters. Using

information from the registrar’s office, the following data

were obtained. Make a control chart and interpret the

result.

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Example 6: P-Chart

Table: The raw date

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Example 6: P-Chart

SOLUTION:

Let us view each student as a binomial trial, where success

is the quality or attribute we wish to study. Success means

the student got an A, and failure is not getting an A. Since

the class size is 60 students each semester, the number of

trials is n = 60.

a) The first step is to use the data to estimate the overall proportion of successes. To do this, we pool the data for all 14 semesters, and use the symbol (not to be confused with ) to designate the pooled proportion of success.

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Example 6: P-Chart

Since the pooled estimate for the proportion of successes

is = 0.175, the estimate for the proportion of failures is

= 1 − = 0.825.

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Example 6: P-Chart

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Example 6: P-Chart

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Example 6: P-Chart

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Example 6: P-Chart

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Example 6: P-Chart

(f) Conclusion: The biology department can tell the dean that the proportion of A’s given in Anatomy and Physiology is in statistical control, with the exception of one unusually good class two semesters ago.