Question 1

Given a standardized score of 1.5, a mean of 40, and a standard deviation of 5, what is the corresponding original score?

\[\begin{eqnarray*} x&=&\frac{x-\mu}{\sigma} \\ 1.5 &=&\frac{x-40}{5} \\ 7.5&=&x-40 \hspace{1cm} \\ 47.5&=&x \hspace{1.98cm} \end{eqnarray*}\]

Question 2

We have a normal distribution of test scores with \(\bar{x} = 75\), and SD = 6.38. What is the proportion of a score falling between a raw score of 81 and 83?

rawscore1 <- pnorm(81, 75, 6.38)
print(rawscore1)
## [1] 0.8265038
rawscore2 <-pnorm(83, 75, 6.38)
print(rawscore2)
## [1] 0.8950642
result <- rawscore2 - rawscore1
print(result)
## [1] 0.06856043

Question 3

You need to score in the top 10% on a statistics test in order to earn a certificate. The class mean is 78 and the variation is 30.25. What score do you need to get the valuable piece of paper?

Let’s write what we have:

  • Mean is 78

  • Variation is 30.25

  • Standard Deviation is 5.5

  • Z-score is 1.28

Note, the top 10% implies that you must score a 90% or greater.

certificatescore1 <- qnorm(.9, 78, 5.5)
print(certificatescore1)
## [1] 85.04853

Question 4

Make a null hypothesis and alternative hypothesis for each statement. Then, indicate if the researcher performs a directional or non-directional test.

  1. There is a difference between high and low income families in the mean amount of assistance families offer to their children in school-related activities

  2. The mean amount of money spent on food by undergraduate student athletes is greater than the mean amount of money spent on food by other graduates.