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 normally distributed. B. No; as long as more than 30 samples are collected, the sample means will be normally distributed. C. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size. D. No; the original population is normally distributed, 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 normally distributed.
B. No; as long as more than 30 samples are collected, the sample means will be normally distributed.
C. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size.
D. No; the original population is normally distributed, 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 the sample means to be normally distributed. B. No; as long as more than 30 samples are collected, the sample means will be normally distributed. C. No; the samples are collected randomly, so the sample means will be normally distributed for any sample size. D. No; the original population is normally distributed, so the sample means will be normally distributed for any sample size.

Solution

Solution Steps

Step 1: Understanding the Distribution of Sample Means

The weights of golden retriever dogs are normally distributed. According to the Central Limit Theorem (CLT), if the original population is normally distributed, the distribution of the sample means will also be normally distributed, regardless of the sample size. Therefore, even with a sample size of $ n = 15 $, the sample means can be treated as being from a normal distribution.

Step 2: Calculating the Probability

We calculated the probability that the sample mean falls within the range from negative infinity to positive infinity. The results are as follows:

The Z-score for the lower bound of the range is $ Z_{start} = -\infty $.
The Z-score for the upper bound of the range is $ Z_{end} = \infty $.

Using the cumulative distribution function $ \Phi $, we find: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-\infty) = 1.0 \] This indicates that the probability that the sample mean falls within the specified range is $ 1.0 $.

Step 3: Conclusion

Since the original population is normally distributed, we conclude that the sample means will also be normally distributed for any sample size, including $ n = 15 $.

Final Answer

The correct answer is $ \boxed{D} $.

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