Question

1. A population has a mean of 40 and a standard deviation of 10. Find the z-scores corresponding to each of the following raw scores:

1. 60.00

2. 32.46

2. A population has a mean of 3 and a standard deviation of 3. Turn the following z scores into raw scores:

1. Z score: 1.75

2. Z score: -2.35

3. For the z-scores below, find the percentile rank (percent of individuals scoring below):

1. 2

2. -0.5

4. First graders in the state of Virginia get an average score of 20 on a reading test (higher score reflect higher levels of performance). A teacher is using a new method to teach reading. She predicts that by the end of the first grade, students getting her new method will have significantly higher scores on reading than those in the population. The mean score of the 25 students in her class is 23.2 and the standard deviation of the population is 4.7.

1. State the null and alternative hypotheses.

2. Calculate the z-score.

Answer 1

1.Population Mean , = 40

Standard Deviation, = 10

a. When X = 60,

Z = (X -   ) / = (60 - 40) / 10 = 2

b. When X = 32.46,

Z = (X -   ) / = ( 32.46 - 40 ) /10 = -0.754

2. Population Mean , = 3

Standard Deviation, = 3

a. Z score = 1.75

Raw score, X = Z * + = 1.75*3 +3 = 8.25

b. Z score: -2.35

Raw score, X = Z * + = -2.35*3 +3 = -4.05

3. a. The percentile rank for Z = 2 is 0.9772 = 97.72%. This value is obtained from Z distribution table . It means that 97.72% of the values lies below the point where Z = 2.

b. The percentile rank for Z = -0.5 is 0.30854 = 30.85%. This value is obtained from Z distribution table . It means that 30.85% of the values lies below the point where Z = -0.5.

4. Population mean, = 20

Population standard Deviation , = 4.7

Sample Size , n = 25

Sample Mean, = 23.2

Sample standard Deviation , s​​​​​​​ = / = 4.7 / = 4.7 / 5 = 0.94

a. Null Hypothesis, H₀ : The population mean & the sample mean are same. i.e =

Alternate Hypothesis, H_a : Sample Mean is greater than Population mean. i.e >

b. The Z score corresponding to X of 20 is given as,

Z = (X - ) / s = ( 20 - 23.2) / 0.94 = -3.4

Probability corresponding to Z value of -3.4 is 0.00034, which means there is a probability of 0.00034 or 0.034% that sample mean will be not be higher than population mean.

Therefore at = 0.05, the Null Hypothesis can be rejected and the Alternate hypothesis is accepted , which states that the sample means is greater than population mean.