suppose that 80% of all 18 year olds play basketball. if a sample size of 4 is taken, find the probability that the sample proportion lies within and including one standard deviation of the population proportion

To find the probability that the sample proportion lies within one standard deviation of the population proportion, we first need to calculate the standard deviation of the sample proportion.

The standard deviation of a sample proportion is given by the formula:

σ = sqrt((p * (1 - p)) / n)

Where:
- σ is the standard deviation of the sample proportion,
- p is the population proportion (0.8 in this case),
- (1 - p) is the complement of the population proportion, and
- n is the sample size (4 in this case).

Using the given values, we can calculate the standard deviation:

σ = sqrt((0.8 * (1 - 0.8)) / 4)
σ = sqrt((0.8 * 0.2) / 4)
σ = sqrt(0.16 / 4)
σ = sqrt(0.04)
σ = 0.2

The standard deviation of the sample proportion is 0.2.

Now, to find the probability that the sample proportion lies within and including one standard deviation of the population proportion, we can use the normal distribution.

Approximately 68% of the values in a normal distribution lie within one standard deviation of the mean. Since the sample proportion is an estimate of the population proportion, we can assume that it follows a normal distribution.

Therefore, the probability of the sample proportion lying within one standard deviation of the population proportion is approximately 68%.

In this case, the probability is 0.68.