Question

An environmental science teacher at a high school with a large population of students wanted to estimate the proportion of students at the school who regularly recycle plastic bottles. The teacher selected a random sample of students at the school to survey. Each selected student went into the teacher’s office, one at a time, and was asked to respond yes or no to the following question.

Do you regularly recycle plastic bottles?

Based on the responses, a 95 percent confidence interval for the proportion of all students at the school who would respond yes to the question was calculated as (0.584, 0.816)

How many students were in the sample selected by the environmental science teacher?

Answer 1

To estimate the proportion of students at the school who regularly recycle plastic bottles, the teacher used a random sample and calculated a 95 percent confidence interval. Using the formula for the confidence interval for proportions, we can determine that the teacher selected approximately 223 students for the survey.

Step-by-step explanation:

To estimate the proportion of students at the school who regularly recycle plastic bottles, the teacher selected a random sample of students and surveyed them one at a time, asking them if they regularly recycle plastic bottles. Based on the responses, a 95 percent confidence interval for the proportion of all students at the school who would respond yes to the question was calculated as (0.584, 0.816).

To find out how many students were in the sample selected by the teacher, we need to use the formula for the confidence interval for proportions:

n = (Z^2 * p * (1-p)) / (E^2)

Where:

n is the sample size,

Z is the Z-score corresponding to the desired level of confidence (in this case, 95 percent),

p is the estimated proportion of students who recycle plastic bottles (we can use the midpoint of the confidence interval, which is (0.584 + 0.816) / 2 = 0.7),

E is the margin of error (in this case, half the width of the confidence interval, which is (0.816 - 0.584) / 2 = 0.116).

Plugging in the values, we get:

n = (1.96^2 * 0.7 * (1-0.7)) / (0.116^2)

n = 2.992 / 0.013456

n ≈ 222.38

Since the sample size must be a whole number, we round up to the nearest whole number.

Therefore, the teacher selected approximately 223 students for the survey.

Answer 2

60 students

Step-by-step explanation:

The confidence interval of a proportion is given by:

`p\pm z*\sqrt{(p*(1-p))/(n) }`

Where 'p' is the proportion of students who responded 'yes', 'z' is the z-score for a 95% confidence interval (which is known to be 1.960), and 'n' is the number of students in the sample.

If the confidence interval is from 0.584 to 0.816, then:

`p=(0.584+0.816)/(2)=0.7 \\0.816-0.584=2*(1.96*\sqrt{(p*(1-p))/(n)}) \\0.116=1.96*\sqrt{(0.7*(1-0.7))/(n)}\\n=16.8966^2*(0.7*0.3)\\n=60\ students`

60 students were in the sample.