In a school district, the average height of a 6th grader is 60 inches, with a standard deviation of 2 inches. Suppose you draw a random sample of 20 students. What is the probability that the average height of the sampled students will be between 59 and 61 inches?

To solve this problem, we will use the concepts of the normal distribution and the Central Limit Theorem, which is applicable here because we're dealing with the average height of a sample.

Step-by-Step Solution

Identify the given information:
- Mean ([tex]\mu[/tex]) of the population: 60 inches
- Standard deviation ([tex]\sigma[/tex]) of the population: 2 inches
- Sample size ([tex]n[/tex]): 20 students
- We're looking for the probability that the sample mean falls between 59 and 61 inches.

Apply the Central Limit Theorem:
The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed, given a sufficiently large sample size, which is typically considered to be [tex]n \geq 30[/tex]. However, since the underlying population is normal or nearly normal, a sample size of 20 is reasonable.
The mean of the sampling distribution of the sample mean ([tex]\mu_{\bar{x}}[/tex]) is the same as the population mean:
[tex]\mu_{\bar{x}} = \mu = 60 \text{ inches}[/tex]
The standard deviation of the sampling distribution of the sample mean, also known as the standard error (SE), is given by:
[tex]\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{2}{\sqrt{20}}[/tex]
Calculating the standard error, we have:
[tex]\sigma_{\bar{x}} = \frac{2}{\sqrt{20}} = 0.447[/tex]

Find the [tex]z[/tex]-scores for 59 inches and 61 inches:
The [tex]z[/tex]-score formula is:
[tex]z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}}[/tex]
- For 59 inches:
  [tex]z_{59} = \frac{59 - 60}{0.447} = -2.24[/tex]
- For 61 inches:
  [tex]z_{61} = \frac{61 - 60}{0.447} = 2.24[/tex]

Calculate the probability between [tex]z_{59} = -2.24[/tex] and [tex]z_{61} = 2.24[/tex]:
Using the standard normal distribution table or a calculator, find the probabilities corresponding to the [tex]z[/tex]-scores:
- [tex]P(Z < -2.24) \approx 0.0125[/tex]
- [tex]P(Z < 2.24) \approx 0.9875[/tex]
Therefore, the probability that a sample mean falls between 59 and 61 inches is:
[tex]P(-2.24 < Z < 2.24) = P(Z < 2.24) - P(Z < -2.24) = 0.9875 - 0.0125 = 0.975[/tex]

Conclusion

The probability that the average height of a randomly selected sample of 20 sixth graders will be between 59 and 61 inches is approximately 97.5%.