Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The executives hire a
statistical consultant and ask her to determine the mean shopping time, , of customers at the supermarkets. The consultant will collect a random sample of
shopping times at the supermarkets and use the mean of these shopping times to estimate . Assuming that the standard deviation of the population of
shopping times at the supermarkets is 27 minutes, what is the minimum sample size she must collect in order for her to be confident that her estimate is
within 3 minutes of ?
Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole
number that satisfies the requirements).
(If necessary, consult a list of formulas.)

Ask by Stanley Turnbull. in the United States

Mar 27,2025

Question

Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The executives hire a
statistical consultant and ask her to determine the mean shopping time, , of customers at the supermarkets. The consultant will collect a random sample of
shopping times at the supermarkets and use the mean of these shopping times to estimate . Assuming that the standard deviation of the population of
shopping times at the supermarkets is 27 minutes, what is the minimum sample size she must collect in order for her to be confident that her estimate is
within 3 minutes of ?
Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole
number that satisfies the requirements).
(If necessary, consult a list of formulas.)

Ask by Stanley Turnbull. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

We are given that the population standard deviation is $$ \sigma = 27 $$ minutes and we want the estimate of $$ \mu $$ to be within $$ E = 3 $$ minutes at the $$ 95\% $$ confidence level. The formula for the margin of error for the mean when the population standard deviation is known is

$$
E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}.
$$

For a $$ 95\% $$ confidence level, we have $$ z_{\alpha/2} = 1.96 $$. We wish to find the minimum sample size $$ n $$ such that

$$
1.96 \frac{27}{\sqrt{n}} \leq 3.
$$

**Step 1: Isolate $$\sqrt{n}$$**

Divide both sides by $$1.96$$:

$$
\frac{27}{\sqrt{n}} \leq \frac{3}{1.96}.
$$

Now, solve for $$\sqrt{n}$$:

$$
\sqrt{n} \geq \frac{27 \times 1.96}{3}.
$$

**Step 2: Compute the right-hand side**

Calculate $$ \frac{27}{3} = 9 $$. Thus,

$$
\sqrt{n} \geq 9 \times 1.96.
$$

Compute $$ 9 \times 1.96 = 17.640 $$ (rounded to three decimal places).

**Step 3: Solve for $$ n $$**

Square both sides:

$$
n \geq (17.640)^2.
$$

Calculate $$ (17.640)^2 \approx 311.1696 $$.

**Step 4: Determine the Minimum Whole Number**

Since $$ n $$ must be a whole number and must be at least as large as 311.1696, we round up to obtain

$$
n = 312.
$$

Thus, the minimum sample size required is $$ 312 $$.
The minimum sample size required is 312.

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