Questions: A recent study reported that 26% of the residents of a particular community lived in poverty. Suppose a random sample of 300 residents of this community is taken. We wish to determine the probability that 30% or more of our sample will be living in poverty. Complete parts (a) and (b) below. a. Before doing any calculations, determine whether this probability is greater than 50% or less than 50%. Why? A. The answer should be less than 50%, because 0.3 is greater than the population proportion of 0.26 and because the sampling distribution is approximately Normal. B. The answer should be greater than 50%, because the resulting z-score will be positive and the sampling distribution is approximately Normal. C. The answer should be greater than 50%, because 0.3 is greater than the population proportion of 0.26 and because the sampling distribution is approximately Normal. D. The answer should be less than 50%, because the resulting z-score will be negative and the sampling distribution is approximately Normal.

A recent study reported that 26% of the residents of a particular community lived in poverty. Suppose a random sample of 300 residents of this community is taken. We wish to determine the probability that 30% or more of our sample will be living in poverty. Complete parts (a) and (b) below. a. Before doing any calculations, determine whether this probability is greater than 50% or less than 50%. Why? A. The answer should be less than 50%, because 0.3 is greater than the population proportion of 0.26 and because the sampling distribution is approximately Normal. B. The answer should be greater than 50%, because the resulting z-score will be positive and the sampling distribution is approximately Normal. C. The answer should be greater than 50%, because 0.3 is greater than the population proportion of 0.26 and because the sampling distribution is approximately Normal. D. The answer should be less than 50%, because the resulting z-score will be negative and the sampling distribution is approximately Normal.
Transcript text: A recent study reported that $26 \%$ of the residents of a particular community lived in poverty. Suppose a random sample of 300 residents of this community is taken. We wish to determine the probability that $30 \%$ or more of our sample will be living in poverty. Complete parts (a) and (b) below. a. Before doing any calculations, determine whether this probability is greater than $50 \%$ or less than $50 \%$. Why? A. The answer should be less than $50 \%$, because 0.3 is greater than the population proportion of 0.26 and because the sampling distribution is approximately Normal. B. The answer should be greater than $50 \%$, because the resulting $z$-score will be positive and the sampling distribution is approximately Normal. C. The answer should be greater than $50 \%$, because 0.3 is greater than the population proportion of 0.26 and because the sampling distribution is approximately Normal. D. The answer should be less than $50 \%$, because the resulting $z$-score will be negative and the samolina distribution is aboroximately Normal.
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Solution

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Solution Steps

Step 1: Calculate the Z-Score

To determine the probability that \(30\%\) or more of the sample will be living in poverty, we first calculate the Z-score using the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

  • \(X = 0.3\) (sample proportion),
  • \(\mu = 0.26\) (population proportion),
  • \(\sigma = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.26(1-0.26)}{300}} \approx 0.0253\).

Substituting the values, we find:

\[ z = \frac{0.3 - 0.26}{0.0253} \approx 1.5795 \]

Step 2: Calculate the Probability

Next, we calculate the probability that the sample proportion is \(30\%\) or more. This is represented as:

\[ P(X \geq 0.3) = P(Z \geq 1.5795) = 1 - P(Z < 1.5795) \]

Using the standard normal distribution, we find:

\[ P(Z < 1.5795) \approx 0.9429 \]

Thus, the probability is:

\[ P(X \geq 0.3) = 1 - 0.9429 = 0.0571 \]

Step 3: Determine the Probability Comparison

Since \(0.0571 < 0.5\), we conclude that the probability of observing \(30\%\) or more of the sample living in poverty is less than \(50\%\).

Final Answer

The answer is A. The answer should be less than \(50\%\), because \(0.3\) is greater than the population proportion of \(0.26\) and because the sampling distribution is approximately Normal.

\(\boxed{A}\)

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