Math  /  Data & Statistics

QuestionThis question: 1 point(s) possib
Given below are the number of successes and sample size for a simple random sample from a population. x=6,n=40,90% level x=6, n=40,90 \% \text { level } 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 pp and express the confidence interval in terms of the sample proportion and the margin of error. a. p^=\hat{p}= \square (Type an integer or a decimal. Do not round.) b. Is the one-proportion z-interval procedure appropriate? Select all that apply. A. The procedure is appropriate because the necessary conditions are satisfied. B. The procedure is not appropriate because nxn-x is less than 5 . C. The procedure is not appropriate because the sample is not à simple random sample. D. The procedure is not appropriate because x is less than 5 . c. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The 90%90 \% confidence interval is from \square to \square . (Round to three decimal places as needed. Use ascending order.) B. The one-proportion z-interval procedure is not appropriate. d. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The margin of error is \square . The 90%90 \% confidence interval is \square ±\pm \square .

Studdy Solution

STEP 1

What is this asking? We're trying to find the proportion of successes in a sample, check if we can use a specific statistical method (the one-proportion z-interval), and if so, use it to calculate a confidence interval and margin of error. Watch out! We need to make sure we meet the conditions for using the one-proportion z-interval, and we need to be careful with our calculations and rounding.

STEP 2

1. Calculate Sample Proportion
2. Check Z-Interval Appropriateness
3. Calculate Confidence Interval
4. Calculate Margin of Error

STEP 3

Let's **dive right in** and calculate the **sample proportion**, p^\hat{p}, which is simply the number of successes (xx) divided by the sample size (nn).
We're given x=6x = 6 and n=40n = 40.

STEP 4

So, p^=xn=640=320=0.15\hat{p} = \frac{x}{n} = \frac{6}{40} = \frac{3}{20} = 0.15.
Our **sample proportion** is p^=0.15\hat{p} = \textbf{0.15}.

STEP 5

To use the one-proportion z-interval, we need to check some conditions.
First, the sample must be a simple random sample, which is stated in the problem.

STEP 6

Next, both np^n\hat{p} and n(1p^)n(1 - \hat{p}) must be greater than or equal to 5.
Let's calculate these values.
We have np^=400.15=6n\hat{p} = 40 \cdot 0.15 = 6, and n(1p^)=40(10.15)=400.85=34n(1 - \hat{p}) = 40 \cdot (1 - 0.15) = 40 \cdot 0.85 = 34.
Since both 6\textbf{6} and 34\textbf{34} are greater than or equal to 5, the conditions are met!

STEP 7

Since the conditions are met, we can calculate the 90% confidence interval.
The formula is p^±zp^(1p^)n\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}, where zz^* is the critical value for a 90% confidence level.

STEP 8

For a 90% confidence level, z=1.645z^* = \textbf{1.645}.
Now, let's plug in our values: 0.15±1.6450.15(10.15)400.15 \pm 1.645 \cdot \sqrt{\frac{0.15(1 - 0.15)}{40}}.

STEP 9

This simplifies to 0.15±1.6450.150.8540=0.15±1.6450.127540=0.15±1.6450.00318750.15±1.6450.0564580.15 \pm 1.645 \cdot \sqrt{\frac{0.15 \cdot 0.85}{40}} = 0.15 \pm 1.645 \cdot \sqrt{\frac{0.1275}{40}} = 0.15 \pm 1.645 \cdot \sqrt{0.0031875} \approx 0.15 \pm 1.645 \cdot 0.056458.

STEP 10

So, the confidence interval is approximately 0.15±0.09290.15 \pm 0.0929, which means the interval is from 0.150.0929=0.0570.15 - 0.0929 = \textbf{0.057} to 0.15+0.0929=0.2430.15 + 0.0929 = \textbf{0.243}.

STEP 11

The **margin of error** is the value we added and subtracted from p^\hat{p} in the previous step.
We already calculated this to be approximately 0.093\textbf{0.093} (rounded to three decimal places).

STEP 12

a. p^=0.15\hat{p} = 0.15 b. A. The procedure is appropriate because the necessary conditions are satisfied. c. A. The 90% confidence interval is from 0.057 to 0.243. d. A. The margin of error is 0.093.
The 90% confidence interval is 0.15 ± 0.093.

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