SOLUTION: Americans ate an average of
25.7 pounds of confectionary products each last year
and spent an average of $61.50 per person doing so. If
the standard deviation for consumption is
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-> SOLUTION: Americans ate an average of
25.7 pounds of confectionary products each last year
and spent an average of $61.50 per person doing so. If
the standard deviation for consumption is
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Question 1161694: Americans ate an average of
25.7 pounds of confectionary products each last year
and spent an average of $61.50 per person doing so. If
the standard deviation for consumption is 3.75 pounds
and the standard deviation for the amount spent is
$5.89, find the following:
a. The probability that the sample mean confectionary
consumption for a random sample of 40 American
consumers was greater than 27 pounds
b. The probability that for a random sample of 50, the
sample mean for confectionary spending exceeded
$60.00 Answer by CPhill(2285) (Show Source):
You can put this solution on YOUR website! ```python
import math
from scipy import stats
# Part a
mu_a = 25.7
sigma_a = 3.75
n_a = 40
x_bar_a = 27
se_a = sigma_a / math.sqrt(n_a)
z_a = (x_bar_a - mu_a) / se_a
p_a = 1 - stats.norm.cdf(z_a)
print(f"Part a:")
print(f" SE = {se_a:.6f}")
print(f" z = {z_a:.6f}")
print(f" P(X_bar > 27) = {p_a:.6f} ({p_a:.4f})")
# Part b
mu_b = 61.50
sigma_b = 5.89
n_b = 50
x_bar_b = 60.00
se_b = sigma_b / math.sqrt(n_b)
z_b = (x_bar_b - mu_b) / se_b
p_b = 1 - stats.norm.cdf(z_b)
print(f"\nPart b:")
print(f" SE = {se_b:.6f}")
print(f" z = {z_b:.6f}")
print(f" P(X_bar > 60.00) = {p_b:.6f} ({p_b:.4f})")
```
```text
Part a:
SE = 0.592927
z = 2.192513
P(X_bar > 27) = 0.014171 (0.0142)
Part b:
SE = 0.832972
z = -1.800781
P(X_bar > 60.00) = 0.964131 (0.9641)
```
### Part a: Probability that sample mean consumption is greater than 27 pounds
**Given:**
* Population mean ($\mu$) = $25.7\text{ lbs}$
* Population standard deviation ($\sigma$) = $3.75\text{ lbs}$
* Sample size ($n$) = $40$
* Target sample mean ($\bar{x}$) = $27\text{ lbs}$
**Step 1: Calculate the Standard Error ($SE$)**
