Question

Determine the point estimate of the population proportion, the margin of error

for the following confidence interval, and the number of individuals in the
sample with the specified characteristic, x, for the sample size provided.
Lower bound = 0.412. upper bound = 0.878. n = 1000

Answer 1

The point estimate of the population proportion is 0.645, the margin of error of the confidence interval is 0.233, and the number of individuals with the specified characteristic x in the sample is 645.

Step-by-step explanation:

To determine the point estimate of the population proportion, we average the lower and upper bounds of the confidence interval. In this case, the confidence interval is given as (0.412, 0.878). Therefore, the point estimate is (0.412 + 0.878) / 2 = 0.645.

To find the margin of error (ME) for the confidence interval, we subtract the point estimate from the upper bound of the confidence interval or subtract the lower bound from the point estimate (since the margin of error is the same on either side of the point estimate). Using the upper bound, ME = 0.878 - 0.645 = 0.233.

To determine the number of individuals in the sample with the specified characteristic, x, we multiply the point estimate by the sample size. So, x = 0.645 * 1000 = 645.

Answer 2

`\hat p=0.645\\\\ME=0.233\\\\x=645 \ individuals`

Step-by-step explanation:

-Given the boundaries as 0.412 and 0.878

-
`\hat p` is the point estimate for the population proportion and is calculated as follows:

`\hat p=(Upper \ Bound+Lower \ Bound)/(2)\\\\=(0.878+0.412)/(2)\\\\=0.645\\\\`

#The margin of error, ME can be calculated for the confidence intervals using the formula:

`ME=(Upper \ Bound-Lower \ Bound)/(2)\\\\=(0.878-0.412)/(2)\\\\=0.233`

#The number of individuals in the sample is the product of the point estimate and population size:

`\hat p=(x)/(n)\\\\x=\hat pn\\\\=0.645* 1000\\\\=645`

Hence, there are 645 individuals in the sample.