Math  /  Data & Statistics

QuestionGiven below are the number of successes and sample size for a simple random sample from a population. x=7,n=50,99% level x=7, n=50,99 \% \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^=0.14\hat{p}=0.14 (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 xx is less than 5 . C. The procedure is not appropriate because nxn-x is less than 5 . D. The procedure is not appropriate because the sample is not a simple ramom sample. c. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The 99%99 \% confidence interval is from .0135 to .2665 . (Round to three decimal places as needed. Use ascending order.) B. The one-proportion z-interval procedure is not appropriate.

Studdy Solution

STEP 1

What is this asking? We're trying to estimate the *true* proportion of successes in a population based on a sample, and we want to be 99% confident that our interval contains the true proportion. Watch out! There are conditions to check before using the one-proportion z-interval procedure!
Make sure the sample is random, and we have enough successes and failures.

STEP 2

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

STEP 3

Let's **dive in** by calculating the sample proportion, p^\hat{p}.
This is simply the number of successes (xx) divided by the sample size (nn).
It's like figuring out what fraction of our little sample was a "win"!

STEP 4

We're given x=7x = \textbf{7} successes and n=50n = \textbf{50} as our sample size.
So, our sample proportion is: p^=xn=750=0.14 \hat{p} = \frac{x}{n} = \frac{\textbf{7}}{\textbf{50}} = \textbf{0.14} So, in our sample, \textbf{14%} were successes.

STEP 5

Before using the z-interval, we need to make sure it's the right tool for the job!
We need to check some conditions.
First, is our sample a simple random sample?
The problem says it is!
Awesome!

STEP 6

Next, we need enough successes and failures.
We need both np^n \cdot \hat{p} and n(1p^)n \cdot (1 - \hat{p}) to be at least 5.
Let's check: np^=500.14=7 n \cdot \hat{p} = 50 \cdot 0.14 = \textbf{7} Since 7 is greater than 5, we're good on successes!
Now for failures: n(1p^)=50(10.14)=500.86=43 n \cdot (1 - \hat{p}) = 50 \cdot (1 - 0.14) = 50 \cdot 0.86 = \textbf{43} 4343 is definitely greater than 5, so we have enough failures too!
The z-interval procedure is appropriate!

STEP 7

For a 99% confidence level, our z-score (zα/2z_{\alpha/2}) is 2.576\textbf{2.576}.
You can find this using a z-table or calculator.
This z-score tells us how many standard deviations away from the mean we need to go to capture 99% of the data in a normal distribution.

STEP 8

Now, let's calculate the standard error, which tells us how much our sample proportion might vary from the true population proportion: SE=p^(1p^)n=0.14(10.14)50=0.140.8650=0.1204500.049 SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.14 \cdot (1 - 0.14)}{50}} = \sqrt{\frac{0.14 \cdot 0.86}{50}} = \sqrt{\frac{0.1204}{50}} \approx \textbf{0.049}

STEP 9

The margin of error (ME) is the z-score times the standard error: ME=zα/2SE=2.5760.0490.126 ME = z_{\alpha/2} \cdot SE = 2.576 \cdot 0.049 \approx \textbf{0.126}

STEP 10

Finally, the confidence interval is our sample proportion plus or minus the margin of error: p^±ME=0.14±0.126 \hat{p} \pm ME = 0.14 \pm 0.126 This gives us a 99% confidence interval of approximately (0.140.126,0.14+0.126)(0.14 - 0.126, 0.14 + 0.126), or (0.014,0.266)(0.014, 0.266).

STEP 11

We already calculated the margin of error in the previous step to be approximately 0.126\textbf{0.126}.
This means we're 99% confident that our sample proportion is within 0.126 of the true population proportion.

STEP 12

a. The sample proportion is p^=0.14\hat{p} = 0.14. b. The one-proportion z-interval procedure *is* appropriate (A). c. The 99% confidence interval is from 0.014 to 0.266.

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