Questions: Weights of golden retriever dogs are normally distributed. Samples of weights of golden retriever dogs, each of size n=15, are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as being from a normal distribution because the sample size is too small? Explain. Choose the correct answer below. A. Yes; the sample size must be over 30 for the sample means to be considered normally distributed. B. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size. C. No; as long as more than 30 samples are collected the sample means will be normally distributed. D. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size.

Weights of golden retriever dogs are normally distributed. Samples of weights of golden retriever dogs, each of size n=15, are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as being from a normal distribution because the sample size is too small? Explain.

Choose the correct answer below.
A. Yes; the sample size must be over 30 for the sample means to be considered normally distributed.
B. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.
C. No; as long as more than 30 samples are collected the sample means will be normally distributed.
D. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size.

Transcript text: Weights of golden retriever dogs are normally distributed. Samples of weights of golden retriever dogs, each of size $n=15$, are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as being from a normal distribution because the sample size is too small? Explain. Choose the correct answer below. A. Yes; the sample size must be over 30 for th ormally distributed B. No; the original population is normally distriutied, so the sample ans will be normally distributed for any sample size C. No; as longg as more than 30 samples are cullected the sampleneans will be normally distributed D. No; the samples are collected randomly, so the sample mea. normally distributed for any sample size.

Solution

Solution Steps

Step 1: Understanding the Problem

We are given that the weights of golden retriever dogs are normally distributed. We need to determine if the sample means from a sample size of $ n = 15 $ can be treated as being from a normal distribution.

Step 2: Applying the Central Limit Theorem

According to the Central Limit Theorem, if the original population is normally distributed, the distribution of the sample means will also be normally distributed regardless of the sample size. Since the population of weights is normally distributed, the sample means will also be normally distributed for any sample size, including $ n = 15 $.

Step 3: Calculating the Probability

To illustrate this, we calculated the probability that the sample mean falls within a specific range. The Z-scores for the range were calculated as follows:

\[ Z_{start} = -0.7746, \quad Z_{end} = 0.7746 \]

Using the cumulative distribution function $ \Phi $, we found:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.7746) - \Phi(-0.7746) = 0.5614 \]

This indicates that there is a $ 56.14\% $ probability that the sample mean falls within the specified range.

Step 4: Conclusion

Based on the above analysis, we conclude that the sample means can indeed be treated as being from a normal distribution because the original population is normally distributed.