The Wechsler Intelligence Scale for Children (Revised) (WISC-R) is a standard intelligence test for children which is widely used in North America. The test was designed so that scores taken from a representative sample of children in a given age group fall in a normal distribution, with a mean of 100 and a standard deviation of 15 {N(100, 15)}.

However, the WISC-R is a time-consuming test which requires extensive training to administer and interpret. You have been hired as a psychologist for an enormous school district in the Canadian North, and one of your tasks is to conduct a massive intelligence screening of the Inuit children, in order to determine which students should be ‘streamed’ into remedial or gifted education programmes.

You cannot test all the children yourself, and you are also concerned that the WISC-R is culturally biased against Inuit children. So you design a simple, short IQ test (called NORTHTEST) which can easily be administered by teachers trained by you in a one-day workshop. Before you use NORTHTEST, you need to standardize it and you need to know how it compares to the WISC-R. You’ll find the data in the PsychLab2-NORTEST.xlsx spreadsheet.

Our data for the NORTHTEST scores is as follows:

lab2 <- c(4,4,6,7,8,10,12,13,13,14,14,14,16,16,16,16,17,17,17,17,17,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,21,21,21,21,21,21,21,22,22,22,22,22,22,22,22,23,23,23,24,24,24,24,24,25,25,25,25,26,26,27,27,27,28,28,29,29,30,31,32,32,34,35,36)

1. You give the test to a representative sample of 85 Inuit children. Using the scores provided (in file NORTEST), determine:

(a) the mean and standard deviation of the scores

We can use the commands mean(lab2) and sd(lab2) on our data to get the mean and standard deviation of our data:

Mean: This is the average test score for the data we are given.

mean(lab2)
## [1] 20.72941

Standard deviation: This is a measure of the spread of the data we are given.

sd(lab2)
## [1] 6.379633

(b) whether these scores fall in a normal distribution (show your work).

We can determine if these scores fall in a normal distribution by inspecting the histogram of our data:

hist(lab2, breaks = 20, main = paste("Histogram of NORTHTEST Scores"), xlab = "Scores")

From this, we can see that our distribution is approximately symmetrical and bell shaped, one exception being that the minimum and maximum scores having a slightly higher frequency than their neighbouring scores. This means we can say our distribution is approximately normal.

2. If you decide that children with a score less than 10 should be placed in a remedial stream, what percentage OR proportion of children will fall into that category? If children with scores greater than 30 should be placed in a gifted stream, what percentage OR proportion of children would fall into that category?

The proportion of children with scores less than 10 is calculated by determining the area under the normal curve under a score of 10. We can do this in R by using the pnorm function. For children with scores less than 10, the lower bound is 0 and the upper bound is 10:

# Setting the lower bound, upper bound mean, and standard deviation:
ub = 10
lb = 0
mean <- mean(lab2)
standardDeviation <- sd(lab2)

#Calculating the proportion of children with scores under 10: 
lessThan10 <- pnorm(ub, mean, standardDeviation) - pnorm(lb, mean, standardDeviation)
lessThan10
## [1] 0.04572319

So the proportion of children with scores less than 10 is 0.046, or approximately 4.6%.

We can use the same method to calculate the proportion of children with scores greater than 30. This time, we will use the pnorm function to find the proportion of children with scores under 30, and then use (1 - the result) to get the proportion of children with scores over 30 (since the area under the normal curve is 1, or 100%).

# Setting the boundary score:
boundaryScore = 30

# Calculating the proportion of children with scores under 10:
greaterThan30 <- 1 - pnorm(boundaryScore, mean, standardDeviation)
greaterThan30
## [1] 0.07309053

So the proportion of children with scores greater than 30 is 0.073, or approximately 7.3%.

3. If the cutoff scores for gifted and remedial students are to be set at 1.5 standard deviations above/below the mean (respectively), then what are the cutoff scores?

The cutoff scores can be calculated by subtracting 1.5(Standard Deviation) from the mean for the lower cutoff (remedial students will be below this score), or adding 1.5(Standard Deviation) to the mean for the higher cutoff(gifted students will be above this score):

# Calculating the cutoff score for remedial students:
remedialCutoff <- mean - 1.5*(standardDeviation)
remedialCutoff
## [1] 11.15996

The cutoff for remedial students is approximately 11.16 points or lower using this method.

# Calculating the cutoff score for gifted students:
giftedCutoff <- mean + 1.5*(standardDeviation)
giftedCutoff
## [1] 30.29886

The cutoff for gifted students is approximately 30.30 points or higher.

4. If a particular child scores 25 on NORTHTEST, what would they score on the WISC-R (assuming that the child would perform consistently)? What would they score on the WISC-R if their score on NORTHTEST was 14?

In this question, we will assume that scoring “consistently” means that if a student scores 1.3 standard deviations higher than the mean on one test, they will also score 1.3 standard deviations higher than the mean on the other test. To compare test scores based on their standard deviations from the mean, we first need to calculate the normalized z score, which is the number of standard deviations away from the mean for a certain score. We then need to multiply this value by the standard deviation of WISC-R scores (15) to find the distance this score will be from the WISC-R mean (100). Finally, we add this number to the WISC-R mean (100) to find the equivalent score on the WISC-R test.

# Calculating the z score of a  child who scores 25 on NORTHTEST:
zScore25 <- (25 - mean)/standardDeviation
zScore25
## [1] 0.6694097
# Calculating the equivalent WISC-R score:
WISC25 <- (zScore25*15) + 100
WISC25
## [1] 110.0411

A student who scores 25 on the NORTHTEST will score approximately 110.04 on the WISC-R (assuming continuous rather than discrete points).

# Calculating the z score of a child who scores 14 on NORTHTEST:
zScore14 <- (14 - mean)/standardDeviation
zScore14
## [1] -1.054827
# Calculating the equivalent WISC-R score:
WISC14 <- (zScore14*15) + 100
WISC14
## [1] 84.17759

A student who scores 14 on the NORTHTEST will score approximately 84.18 on the WISC-R.

5. Based on what you now know about NORTHTEST, is it a suitable substitute for the WISC-R? Why or why not? (Hints: Can scores on NORTHTEST be fairly compared with scores on the WISC-R? Is NORTHTEST a valid measure of the IQ of Inuit children? Could NORTHTEST be used for other North American children? etc.)

NORTHTEST is likely not a suitable substitute for the WISC-R. Even though the scores of both tests fall into approximately normal distributions, there is no guarantee that they are measuring the same skills. For example, the NORTHTEST could have many numerical questions compared to the WISC-R. A student who performs poorly on numerical questions may receive a low score on the NORTHTEST, and a much higher score on the WISC-R (compared using standard deviations, as in question 4). It is highly unlikely that NORTHTEST weights different skills in exactly the same way as the WISC-R, so the scores can not be fairly compared.

We also know that the WISC-R is time consuming to teach someone to administer, time consuming to administer, and time consuming to interpret. On the other hand, NORTHTEST is not time consuming in any of these areas. The WISC-R was probably designed to eliminate as much variation in test-giving procedures as possible, so no student has an advantage due to how the test was administered, and test results are interpreted in the same way for every student. Since NORTHTEST spends far less time on these things, there will likely be more variation in how the test is administered and interpreted, leading to some students possibly getting lower or higher scores than they should.

If the NORTHTEST questions are tailored to Inuit children, it is likely that other North American children would not perform as well on the NORTHTEST, not because they have lower IQ, but simply because of cultural differences. In trying to create a test that is not biased against Inuit students, the test may be biased against other North American cultures.