Question

In each of Exercises$11.25-11.30$, we have given the number of successes and the sample size for a simple random sample from a population. In each case, do the following tasks.
a. Determine the sample proportion.
b. Decide whether using the one-proportion$z$-interval procedure is appropriate.
c. If appropriate, use the one-proportion$z$-interval procedure to find the confidence interval at the specified confidence level.
d. If appropriate, find the margin of error for the estimate of $p$and express the confidence interval in terms of the sample proportion and the margin of error:

11.29 $x=16,n=20,90\%$level

Short answer

(a) The sample proportion is $0.8$.

(b) The one proportion $z$-test interval approach is not appropriate.

(c) The confidence interval cannot be calculated. Because,$z$test is not appropriate.

(b) The margin of error cannot be calculated. Because, $z$test is not appropriate.

Step-by-step solution

Part (a) Step 1: Given information

To determine the sample proportion for $x=16,n=20,90\%$ level.

Part (a) Step 2: Explanation

Let, the sample size $n$is $20$
And the number of successes $x$is $16$.
Determine the sample proportion by:
$\hat{p}=\frac{x}{n}$
$=\frac{16}{20}$
$=0.8$
Asa a result, the sample proportion is $0.8$.

Part (b) Step 1: Given information

To decide whether using the one-proportion $z$-interval procedure is appropriate.

Part (b) Step 2: Explanation

The following are the assumptions for one proportion $z$-interval procedure:

A simple random sampling method should be used to select the sample.
The number of successes $x$and failures$n-x$ should both be at least five.
The number of successes in this case, $x=16$, is larger than $5$.
The following formula is used to calculate the number of failures:
$n-x=20-16$
$=4$

The number of failures is less than $5$in this case.

As a result, the one proportion z-test interval approach is not appropriate.

Part (c) Step 1: Given information

To appropriate, use the one-proportion $z$-interval procedure to find the confidence interval at the specified confidence level.

Part (c) Step 2: Explanation

It is clear from part (b) that the one-proportion $z$-interval procedure is ineffective.
As a result, the confidence interval cannot be calculated.

Part (d) Step 1: Given information

To appropriate, find the margin of error for the estimate of $p$and express the confidence interval in terms of the sample proportion and the margin of error:

Part (d) Step 2: Explanation

It is clear from part (b) that the one-proportion $z$ interval procedure is ineffective.
As a result, the margin of error cannot be calculated.