Question

Patients recovering from a stroke have their grip strengths measured in each hand to monitor their progress. A certain population of over 100 male patients have grip strengths in their dominant hands with a mean and standard deviation of approximately 41 kg and 9 kg, respectively. Suppose that we take random samples of 4 male patients from this population and calculate bar(x) as the sample mean grip strength from each group of patients. What will be the shape of the sampling distribution of bar(x)? 1) Normal distribution 2) Uniform distribution 3) Exponential distribution 4) Binomial distribution

Answer 1

The shape of the sampling distribution of the sample mean grip strength will be a Normal distribution.

Step-by-step explanation:

The shape of the sampling distribution of the sample mean grip strength will be a Normal distribution.

According to the Central Limit Theorem, when the sample size is large enough, the sampling distribution of the sample mean will approximate a normal distribution regardless of the shape of the population distribution. Since the population size is larger than 100 and the sample size is 4, we can assume that the sample mean grip strength will follow a normal distribution.

This means that the sample means from multiple groups of patients will cluster around the population mean grip strength of 41 kg, and the distribution of these sample means will be symmetric and bell-shaped.

Answer 2

The shape of the sampling distribution of the sample mean
`\( \bar{x} \)` will be approximately normally distributed due to the Central Limit Theorem (CLT). The CLT states that if you take sufficiently large random samples from a population with a finite level of variance, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution.

Given that the population has a known mean and standard deviation, and assuming that the population of grip strengths is roughly bell-shaped (since most biological measurements are normally distributed), the sampling distribution of the sample mean for samples of size 4 will be approximately normal, especially if the original population distribution is normal. This is true even though the sample size of 4 is quite small; the normality assumption for the underlying population distribution aids in justifying the normal shape of the sampling distribution of
`\( \bar{x} \)`.

Therefore, the correct answer is:

1) Normal distribution.