Question

1f. Compare 1a and 1d, and 1b and 1e, explain why the percentage in 1a is much larger than that in 1d and why the value in 1b is much smaller than that in 1e?

1. Suppose that for Edwardsville High School, distances between students’ homes and the high school observe normal distribution with the average distance being 4.76 miles and the standard deviation being 1.74 miles. Express distances and z scores to two decimal places. Write the formula to be used before each calculation.

1a. What percentage of students in the high school live farther than 6.78 miles from the school?

1b. A survey shows that 8% of the students who live closest to the school choose to walk to school. What is the maximum walking distance of these 8% of students? In other words, what is the distance below which these 8% of students live from the school?

                                                                                                                                   

1c. Suppose that the school district’s policy allows students living beyond 4.50 miles from the school to take school buses to go to school. There are 3,567 students who enroll in the fall semester, 2006. How many students in the high school are not eligible to take school buses?

1d. Suppose all samples of size 12 are taken. What percentage of sample means has a value larger than 6.78 miles?

1e. Below what value are 8% of sample means of size 12?

Answer 1

1. Suppose that for Edwardsville High School, distances between students’ homes and the high school observe normal distribution with the average distance being 4.76 miles and the standard deviation being 1.74 miles.

Let X be the distance between the home and the high school of a randomly selected student. We can say that X is normally distributed with mean and standard deviation

1a. What percentage of students in the high school live farther than 6.78 miles from the school?

The probability of a randomly selected student in the high school live farther than 6.78 miles from the school is

ans: The percentage of students in the high school live farther than 6.78 miles from the school is 12.30%

1b. A survey shows that 8% of the students who live closest to the school choose to walk to school. What is the maximum walking distance of these 8% of students? In other words, what is the distance below which these 8% of students live from the school?

Let q miles be the distance below which these 8% of students live from the school. This is same as the probability that a randomly selected student lives less than q miles from the school is 0.08

That is,

In terms of the z values, we need

However, we know that the probability P(Z<0)=0.5. Since we need a probability less than 0.5, the value of z must be negative.

Hence,

Using the standard normal tables, we get for z=1.41, P(Z<1.41)=0.92.

Hence

we need

We can equate the z score of q to -1.41 and get

ans: the distance below which these 8% of students live from the school is 2.31 miles

1c. Suppose that the school district’s policy allows students living beyond 4.50 miles from the school to take school buses to go to school. There are 3,567 students who enroll in the fall semester, 2006. How many students in the high school are not eligible to take school buses?

The probability that a randomly selected student lives beyond 4.5 miles from the school is

Let Y be the number out of 3,567 students who are eligible to take school buses. We can say that Y has a Binomial distribution with parameters, number of trials (number of students in the school) n=3567, and success probability (The probability that a randomly selected student lives beyond 4.5 miles from the school) p=0.5596

The expected value of Y is (using the formula for Binomial distribution)

the number out of 3,567 students who are not eligible to take school buses is

3567 - 1996.16=1570.84

ans: The number of students in the high school who are not eligible to take school buses is 1570.84

1d. Suppose all samples of size 12 are taken. What percentage of sample means has a value larger than 6.78 miles?

Let be the average distance a sample of n=12 students live. Since X has a normal distribution and we know the population standard deviation, we can say that has a normal distribution with mean and standard deviation

The probability that a randomly selected sample of size 12 has a mean has a value larger than 6.78 is

ans: The percentage of sample means has a value larger than 6.78 miles is 0.00%

1e. Below what value are 8% of sample means of size 12?

Let 8% of sample means of size 12 be below q miles. This is same as the probability that a randomly selected sample of 12 has a mean less than q is 0.08

That is,

In terms of the z values, we need

However, we know that the probability P(Z<0)=0.5. Since we need a probability less than 0.5, the value of z must be negative.

Hence,

Using the standard normal tables, we get for z=1.41, P(Z<1.41)=0.92.

Hence

we need

We can equate the z score of q to -1.41 and get

ans: 8% of sample means of size 12 are below 4.05 miles

1f. Compare 1a and 1d, and 1b and 1e, explain why the percentage in 1a is much larger than that in 1d and why the value in 1b is much smaller than that in 1e?

ans: We can see that the standard deviation of the random variable X used in 1a/1b is larger than the standard deviation of variable used in 1d/1e. This indicates that the distribution of sample means for a sample of 12 students has less spread than the distribution for the single student. That means, the sample means are more likely to be closer to the mean 4.76 compared to the value for a single student, making the extreme values of sample means are less likely.

Because of which, the percentage of students who live beyond 6.78 would be larger than the percentage of samples which have a mean greater than 6.78. Hence the percentage in 1a is much larger than that in 1d.

For the same reason,  the distance below which 8% of students live from the school is smaller (individual students are away from the mean 4.76) than the average distance below which 8% of sample means lie (sample means are closer to the mean, 4.76). Hence, the value in 1b is much smaller than that in 1e