4.39 Weights of pennies: The distribution of weights of United States pennies is approximately normal
with a mean of 2.5 grams and a standard deviation of 0.03 grams.
(a) What is the probability that a randomly chosen penny weighs less than 2.4 grams?
(please round to four decimal places)
(b) Describe the sampling distribution of the mean weight of 10 randomly chosen pennies.
Mean: grams (please round to one decimal place)
Standard Deviation:
© What is the probability that the mean weight of 10 pennies is less than 2.4 grams?
(please round to four decimal places)

Question

UpStudy AI · Accepted Answer

$$ \textbf{(a)} $$

The weight $$X$$ of a penny is normally distributed as  
$$
X \sim N(\mu = 2.5, \sigma = 0.03).
$$

We want to find  
$$
P(X < 2.4).
$$

Compute the $$z$$-score for $$X = 2.4$$:
$$
z = \frac{2.4 - 2.5}{0.03} = \frac{-0.1}{0.03} \approx -3.33.
$$

Using the standard normal distribution table or a calculator, we have
$$
P(Z < -3.33) \approx 0.0004.
$$

Thus,  
$$
\boxed{0.0004}.
$$

$$ \textbf{(b)} $$

For a sample of $$ n = 10 $$ pennies, the sampling distribution of the sample mean $$\bar{X}$$ is also normally distributed. Its parameters are as follows:

- Mean:
  $$
  \mu_{\bar{X}} = \mu = 2.5 \text{ grams.}
  $$
  
- Standard deviation (standard error):
  $$
  \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{0.03}{\sqrt{10}} \approx 0.0095 \text{ grams.}
  $$

After rounding the mean to one decimal place, we have:
$$
\text{Mean } = 2.5 \text{ grams, and Standard Deviation } \approx 0.0095 \text{ grams.}
$$

$$ \textbf{(c)} $$

We need the probability that the mean weight of 10 pennies is less than 2.4 grams:
$$
P(\bar{X} < 2.4).
$$

Compute the $$z$$-score for $$\bar{X} = 2.4$$:
$$
z = \frac{2.4 - 2.5}{\sigma_{\bar{X}}} = \frac{-0.1}{0.0095} \approx -10.54.
$$

For such a negative $$z$$-score, the probability is extremely small. From the standard normal distribution,
$$
P(Z < -10.54) \approx 0.0000.
$$

Thus,  
$$
\boxed{0.0000}.
$$
$$ \textbf{(a)} $$ The probability that a randomly chosen penny weighs less than 2.4 grams is 0.0004. $$ \textbf{(b)} $$ The sampling distribution of the mean weight of 10 pennies has a mean of 2.5 grams and a standard deviation of approximately 0.0095 grams. $$ \textbf{(c)} $$ The probability that the mean weight of 10 pennies is less than 2.4 grams is 0.0000.

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