Question

The lengths of adult males' hands are normally distributed with mean 188 mm and standard devlation 157.3 mm. Suppose that 6 individuals are randomly chosen. Round all answers to 4 where possible. What is the distribution of xˉ?

Answer 1

The distribution of sample means (
`\(\bar{x}\)`) for the hand lengths of adult males, drawn from a population with a mean of 188 mm and a standard deviation of 157.3 mm, is normally distributed. The mean of this distribution is also 188 mm, while the standard deviation is approximately 64.18 mm, calculated based on a sample size of 6 individuals.

Step-by-step explanation:

The distribution of sample means
`(\(\bar{x}\))` provides insights into the variability expected when repeatedly drawing samples from a population. In this case, we consider the hand lengths of adult males, assuming a normal distribution in the population with a mean
`(\(\mu\))` of 188 mm and a standard deviation (
`\(\sigma\)`) of 157.3 mm. When six individuals are randomly chosen, the central limit theorem allows us to assert that the distribution of sample means (
`\(\bar{x}\)`) will also be normal, regardless of the population's distribution.

Mathematically, this can be expressed as:

`\[\mu_{\bar{x}} = \mu\]`

`\[\sigma_{\bar{x}} = (\sigma)/(√(n))\]`

The mean of the sample means
`(\(\mu_{\bar{x}}\)`) equals the population mean (
`\(\mu\)`), resulting in a
`\(\mu_{\bar{x}}\)` of 188 mm. The standard deviation of the sample means
`(\(\sigma_{\bar{x}}\))` is determined by dividing the population standard deviation
`(\(\sigma\))` by the square root of the sample size (n), which, in this case, is 6. Consequently,
`\(\sigma_{\bar{x}}\)` is approximately 64.18 mm.

`\[\mu_{\bar{x}} = 188\]`

`\[\sigma_{\bar{x}} = (157.3)/(√(6)) \approx (157.3)/(2.4495) \approx 64.18\]`

This implies that, on average, the sample mean (
`\(\bar{x}\)`) of hand lengths for groups of six adult males will hover around 188 mm, and the variability in these sample means will be reduced compared to individual hand lengths due to the larger sample size. This understanding of the distribution of sample means is fundamental in statistical inference, particularly in constructing confidence intervals and conducting hypothesis tests.

So, the distribution of
`\(\bar{x}\)` has a mean
`(\(\mu_{\bar{x}}\))` of 188 mm and a standard deviation
`(\(\sigma_{\bar{x}}\))` of approximately 64.18 mm.