We are tasked with finding the probability that the sample proportion $\hat{p}$ of households spending more than $125 a week is less than 0.27, given that the population proportion $p = 0.26$ and the sample size $n = 191$. We are to assume that the sampling distribution of $\hat{p}$ follows a normal distribution because the conditions for the normal approximation have been met.

Step 1: Determine the standard error of the sample proportion

The standard error (SE) of the sample proportion $\hat{p}$ is calculated using the formula:

$SE = \sqrt{\frac{p(1 - p)}{n}}$

Where:

• $p = 0.26$ is the population proportion,
• $n = 191$ is the sample size.

Substitute the values into the formula:

$SE = \sqrt{\frac{0.26(1 - 0.26)}{191}} = \sqrt{\frac{0.26 \times 0.74}{191}} = \sqrt{\frac{0.1924}{191}} \approx \sqrt{0.00100785} \approx 0.0317$

Step 2: Find the Z-score

Next, we calculate the Z-score for the sample proportion $\hat{p} = 0.27$ using the formula:

$Z = \frac{\hat{p} - p}{SE}$

Substitute the known values:

$Z = \frac{0.27 - 0.26}{0.0317} = \frac{0.01}{0.0317} \approx 0.3155$

Step 3: Find the probability

We now use the standard normal distribution to find the probability that the Z-score is less than 0.3155. Using a Z-table or standard normal distribution calculator, the probability corresponding to $Z = 0.3155$ is approximately:

$P(Z < 0.3155) \approx 0.6239$

Thus, the probability that the sample proportion is less than 0.27 is approximately 0.6239.

Final Answer:

There is a 0.6239 probability that the sample proportion of households spending more than $125 a week is less than 0.27.

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Do you need any further details or explanations? Here are 5 related questions to expand your understanding:

1. How would the probability change if the sample size was increased?
2. What if the population proportion was different, say 0.30? How would the calculation differ?
3. Why do we use the normal approximation in this case, and what are the conditions for it to be valid?
4. How can we calculate the probability that the sample proportion is greater than 0.27?
5. How would you interpret the result in the context of grocery spending?

Tip: The larger the sample size, the smaller the standard error, which results in a more accurate estimate of the population proportion.