To answer this question, we need to use the Central Limit Theorem (CLT), which states that if we take a large number of random samples from a population, the distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution.

First, we need to calculate the standard error (SE), which is the standard deviation of the sampling distribution of the sample mean. The formula for the standard error is:

SE = σ / sqrt(n)

where σ is the population standard deviation and n is the sample size. Substituting the given values:

SE = 6.4 / sqrt(25) = 1.28

Next, we calculate the z-scores for the lower and upper bounds of the weight range (82.44 grams and 83.72 grams). The formula for a z-score is:

Z = (X - μ) / SE

where X is the value from the dataset, μ is the population mean, and SE is the standard error.

For X = 82.44 grams:

Z1 = (82.44 - 85) / 1.28 = -2

For X = 83.72 grams:

Z2 = (83.72 - 85) / 1.28 = -1

Now, we need to find the area under the standard normal curve between these two z-scores. This can be done using a standard normal distribution table or a statistical software or calculator. The values corresponding to Z = -2 and Z = -1 in the standard normal distribution table are 0.0228 and 0.1587 respectively.

The percentage of samples with a mean weight between 82.44 grams and 83.72 grams is the difference between these two areas:

Percentage = (0.1587 - 0.0228) * 100 = 13.59%

So, we would expect approximately 13.59% of the samples to have a sample mean weight between 82.44 grams and 83.72 grams.

Question

To answer this question, we need to use the Central Limit Theorem (CLT), which states that if we take a large number of random samples from a population, the distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution.

First, we need to calculate the standard error (SE), which is the standard deviation of the sampling distribution of the sample mean. The formula for the standard error is:

SE = σ / sqrt(n)

where σ is the population standard deviation and n is the sample size. Substituting the given values:

SE = 6.4 / sqrt(25) = 1.28

Next, we calculate the z-scores for the lower and upper bounds of the weight range (82.44 grams and 83.72 grams). The formula for a z-score is:

Z = (X - μ) / SE

where X is the value from the dataset, μ is the population mean, and SE is the standard error.

For X = 82.44 grams:

Z1 = (82.44 - 85) / 1.28 = -2

For X = 83.72 grams:

Z2 = (83.72 - 85) / 1.28 = -1

Now, we need to find the area under the standard normal curve between these two z-scores. This can be done using a standard normal distribution table or a statistical software or calculator. The values corresponding to Z = -2 and Z = -1 in the standard normal distribution table are 0.0228 and 0.1587 respectively.

The percentage of samples with a mean weight between 82.44 grams and 83.72 grams is the difference between these two areas:

Percentage = (0.1587 - 0.0228) * 100 = 13.59%

So, we would expect approximately 13.59% of the samples to have a sample mean weight between 82.44 grams and 83.72 grams.

Knowee AI · Accepted Answer