Questions: Question 8 of 13, Step 1 of 1 Correct 3 The mean cost of a five pound bag of shrimp is 47 dollars with a variance of 36. If a sample of 43 bags of shrimp is randomly selected, what is the probability that the sample mean would differ from the true mean by less than 1.4 dollars? Round your answer to four decimal places. Answer How to enter your answer (opens in new window) Tables Keypad Keyboard Shortcuts Submit Answer 2024 Hawkes Learning

Question 8 of 13, Step 1 of 1
Correct
3

The mean cost of a five pound bag of shrimp is 47 dollars with a variance of 36.
If a sample of 43 bags of shrimp is randomly selected, what is the probability that the sample mean would differ from the true mean by less than 1.4 dollars? Round your answer to four decimal places.

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Transcript text: Question 8 of 13, Step 1 of 1 Correct 3 The mean cost of a five pound bag of shrimp is 47 dollars with a variance of 36. If a sample of 43 bags of shrimp is randomly selected, what is the probability that the sample mean would differ from the true mean by less than 1.4 dollars? Round your answer to four decimal places. Answer How to enter your answer (opens in new window) Tables Keypad Keyboard Shortcuts Submit Answer @ 2024 Hawkes Learning

Solution

Solution Steps

Step 1: Given Information

We are given the following parameters for the cost of a five-pound bag of shrimp:

Mean (\( \mu \)): 47 dollars
Variance (\( \sigma^2 \)): 36
Sample size (\( n \)): 43 bags

Step 2: Calculate Standard Deviation

The standard deviation (\( \sigma \)) is calculated as: \[ \sigma = \sqrt{\text{variance}} = \sqrt{36} = 6 \]

Step 3: Define the Range for Sample Mean

We want to find the probability that the sample mean differs from the true mean by less than 1.4 dollars. Therefore, we define the range: \[ \text{Range} = (\mu - 1.4, \mu + 1.4) = (47 - 1.4, 47 + 1.4) = (45.6, 48.4) \]

Step 4: Calculate Z-scores

To find the Z-scores corresponding to the range, we use the formula: \[ Z = \frac{X - \mu}{\sigma / \sqrt{n}} \] Calculating the Z-scores:

For the lower bound (\( X = 45.6 \)): \[ Z_{start} = \frac{45.6 - 47}{6 / \sqrt{43}} \approx -1.5301 \]
For the upper bound (\( X = 48.4 \)): \[ Z_{end} = \frac{48.4 - 47}{6 / \sqrt{43}} \approx 1.5301 \]

Step 5: Calculate the Probability

The probability that the sample mean falls within the specified range is given by: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.5301) - \Phi(-1.5301) \] From the output, we find: \[ P \approx 0.874 \]