Question

1. the distribution of 1st nine weeks grades in statistics is normally distributed with a mean of 60 and a standard deviation of 7. what proportion of grades are above a 67? a.).1784 c.).8413 b.).1587 d.).9221 8. the distribution of 1st nine weeks grades in statistics is normally distributed with a mean of 60 and a standard deviation of 7. what proportion of grades are below 63.1? a.).3289 c.).6711 b.).4255 d.).5755 9. the distribution of 1st nine weeks grades in statistics is normally distributed with a mean of 60 and a standard deviation of 7. what is the 85th percentile of student grades? a.) 60.54 c.) 54.67 b.) 67.26 d.) 49.21 10. the distribution of 1st nine weeks grades in statistics is normally distributed with a mean of 60 and a standard deviation of 7. what grade is 2 standard deviations above the mean? a.) 67 c.) 81 b.) 74 d.) 53 11. the distribution of 1st nine weeks grades in statistics is normally distributed with a mean of 60 and a standard deviation of 7. what grade has a z score of 2? a.) 67 c.) 81 b.) 74 d.) 53

Explanation:

Step1: Calculate the z - score for \(x = 67\)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 60\) (mean) and \(\sigma=7\) (standard deviation).
Substitute \(x = 67\), \(\mu = 60\), and \(\sigma = 7\) into the formula:
\(z=\frac{67 - 60}{7}=\frac{7}{7}=1\)

Step2: Find the proportion above \(z = 1\)

Using the standard normal distribution table, \(P(Z<1)=0.8413\).
Since \(P(Z > z)=1 - P(Z < z)\), then \(P(Z>1)=1 - 0.8413 = 0.1587\)

Step3: Calculate the z - score for \(x = 63.1\)

Substitute \(x = 63.1\), \(\mu = 60\), and \(\sigma = 7\) into \(z=\frac{x-\mu}{\sigma}\):
\(z=\frac{63.1-60}{7}=\frac{3.1}{7}\approx0.44\)
Using the standard normal distribution table, \(P(Z < 0.44)=0.6700\approx0.6711\)

Step4: Find the z - score for the 85th percentile

Looking up the z - score in the standard normal table for \(P(Z<z)=0.85\), \(z\approx1.04\)
Using the formula \(x=\mu+z\sigma\), substitute \(\mu = 60\), \(z = 1.04\), and \(\sigma = 7\):
\(x=60+(1.04\times7)=60 + 7.28=67.28\approx67.26\)

Step5: Calculate the grade 2 standard deviations above the mean

Using the formula \(x=\mu + z\sigma\), with \(z = 2\), \(\mu = 60\), and \(\sigma = 7\)
\(x=60+(2\times7)=60 + 14=74\)

Step6: Calculate the grade with \(z = 2\)

Using \(x=\mu+z\sigma\), with \(z = 2\), \(\mu = 60\), and \(\sigma = 7\)
\(x=60+(2\times7)=60 + 14=74\)

Answer:

1. B. \(0.1587\)
2. C. \(0.6711\)
3. B. \(67.26\)
4. B. \(74\)
5. B. \(74\)