1. A simple random sample of 400 students is taken at a large university. The average height of the sampled students is 68 inches and the SD is 2 inches. The distribution of heights in the sample follows the normal curve very closely.
An approximate 68% confidence for the average height of students at the university is 68 inches plus or minus ____________ inches.
Approximately 68% of the students in the sample have heights in the range 68 inches plus or minus ____________ inches.
To find the approximate 68% confidence interval for the average height of students at the university, we will use the standard deviation (SD) and the sample size.
Since the distribution of heights in the sample follows the normal curve, we can use the concept of the standard error, which is calculated by dividing the standard deviation by the square root of the sample size.
The formula for the standard error (SE) is as follows:
SE = SD / √(n)
where SD is the standard deviation and n is the sample size.
In this case, the standard deviation (SD) is given as 2 inches, and the sample size (n) is 400 students.
Let's plug in these values into the formula:
SE = 2 / √(400)
Calculating the square root of 400, we get:
SE = 2 / 20 = 0.1
The 68% confidence interval can be found by multiplying the standard error by the appropriate value from the standard normal distribution. In this case, since we want the middle 68% of the distribution, we will use the z-score corresponding to 0.5 - (0.68/2) = 0.16.
Consulting a standard normal distribution table or calculator, we find that the z-score corresponding to 0.16 is approximately 0.994.
Finally, we can calculate the margin of error (MOE) by multiplying the standard error by the z-score:
MOE = SE * Z
MOE = 0.1 * 0.994 ≈ 0.0994
Therefore, the approximate 68% confidence for the average height of students at the university is 68 inches plus or minus 0.0994 inches.
For the second part of the question, we can use the margin of error to determine the range within which approximately 68% of the students' heights in the sample fall.
To find the range, we subtract and add the margin of error to the average height:
Lower limit = 68 - 0.0994 = 67.9006
Upper limit = 68 + 0.0994 = 68.0994
Therefore, approximately 68% of the students in the sample have heights in the range of 67.9006 to 68.0994 inches.