Question

Chuckies Choice Chewy Chunky Cheddar Cheese Charcoal Charred Chips" have a mean weight of 5 ounces with a standard deviation of 0.21 ounces. A sample of 40 chips was taken. Find the probability that the sample mean weight is between 4.8 and 5.2 ounces.

Answer 1

The probability that the sample mean weight of Chuckies Choice Chewy Chunky Cheddar Cheese Charcoal Charred Chips falls between 4.8 and 5.2 ounces is approximately 0.9819 or 98.19%.

Step-by-step explanation:

To calculate the probability, we use the z-score formula:
`\(z = \frac{{\bar{X} \mu}}{{(\sigma)/(√(n))}}\),` where
`\(\bar{X}\)` is the sample mean, \(\mu\) is the population mean (5 ounces),
`\(\sigma\)` is the population standard deviation (0.21 ounces), and \(n\) is the sample size (40).

1. Calculate the z-scores for 4.8 and 5.2 ounces:

`\[z_(4.8) = \frac{{4.8 - 5}}{{(0.21)/(√(40))}} \approx -2.357\]`

`\[z_(5.2) = \frac{{5.2 - 5}}{{(0.21)/(√(40))}} \approx 2.357\]`

2. Refer to the standard normal distribution table to find the probabilities associated with these z-scores:

For
`\(z = -2.357\)`, the probability is approximately 0.0092.

For
`\(z = 2.357\)`, the probability is approximately 0.9911.

3. Calculate the probability that the sample mean weight is between 4.8 and 5.2 ounces:

`\[P(4.8 < \bar{X} < 5.2) = P(z_(5.2)) - P(z_(4.8)) \approx 0.9911 - 0.0092 \approx 0.9819\]`

Therefore, the probability that the sample mean weight is between 4.8 and 5.2 ounces is approximately 0.9819 or 98.19%. This indicates a high likelihood that a sample of 40 chips will have a mean weight within this specified range, considering the characteristics of Chuckies Choice Chewy Chunky Cheddar Cheese Charcoal Charred Chips' weights and the provided population parameters.