Exam 1 Review

Chapters 1 - 4

Dave Brocker

Farmingdale State College

Learning Goals

  • Compute sample mean and standard deviation by hand.
  • Interpret what the mean and SD tell us about data.
  • Standardize raw scores into Z-scores.
  • Interpret Z-scores (direction and distance from mean).
  • Relate Z-scores to probability under a normal curve.

1. Mean Calculation

Sample Mean Calculation

A psychology researcher collects the following data on the number of hours five participants spent practicing a task: 3, 5, 8, 2, 7.

Question: What is the average number of hours spent practicing?

\(\bar{x} = \frac{\sum(3+5+8+2+7)}{5} = \frac{25}{5} = 5\text{ hours}\) 

mean(c(3,5,8,2,7))
[1] 5

2. Mean Calculation

Sample Mean Calculation

A study measures the changes in mood scores before and after an intervention. The changes in mood scores are: -4, -2, 0, 3, 5.

Question: Calculate the mean change in mood score.

\(\bar{x} = \frac{\sum(-4 + (-2) + 0 + 3 + 5)}{5} = \frac{2}{5} = 0.4\) 

mean(c(-4,-2,0,3,5))
[1] 0.4

3. Standard Deviation Calculation

Sample Standard Deviation

A psychologist measures the number of stressful events experienced by a small group of participants over a week: 2, 3, 6, 4, 5.

Question: Calculate the standard deviation of stressful events.

  • Mean: \(\bar{x} = \frac{2+3+6+4+5}{5} = 4\)
  • Sum of squares: \((2-4)^2+(3-4)^2+(6-4)^2+(4-4)^2+(5-4)^2 = 4+1+4+0+1 = 10\)
  • \(s = \sqrt{\frac{10}{5-1}} = \sqrt{2.5} \approx 1.58\)
sd(c(2,3,6,4,5))
[1] 1.581139

4. Standard Deviation Calculation

Sample Standard Deviation

Sample Standard Deviation Participants’ anxiety scores in a clinical trial are recorded as: 4, 7, 3, 5, 8, 6, 2.

Question: Compute the standard deviation of these anxiety scores.

  • Mean: \(\bar{x} = \frac{4+7+3+5+8+6+2}{7} = \frac{35}{7} = 5\)
  • Sum of squares: \(1+4+4+0+9+1+9 = 28\)
  • \(s = \sqrt{\frac{28}{7-1}} = \sqrt{4.67} \approx 2.16\)
sd(c(4,7,3,5,8,6,2))
[1] 2.160247

5. Standard Deviation Calculation

Interpreting Standard Deviation in Psychology

A researcher finds that the standard deviation of reaction times in a cognitive task is 1.5 seconds. Another researcher finds a standard deviation of 0.5 seconds for a similar task.

Question: Which task shows more variability in participants’ reaction times, and what might this indicate about task difficulty?

Solution

A standard deviation of 1.5 is is larger than 0.5 is, so the first task shows more variability.

6. Z-Score Calculation

Z-Score for a Single Value

In a personality study, the average score on an extraversion scale is 50, with a standard deviation of 10. One participant scores 70.

Question: What is the Z-score for this participant?

\(z = \frac{X - \mu}{\sigma} = \frac{70 - 50}{10} = \frac{20}{10} = 2\) 

(70-50)/10
[1] 2

7. Z-Score Calculation:

Interpreting Z-Scores

A participant in a self-esteem survey has a Z-score of -1.5. The mean self-esteem score for the survey is 40, and the standard deviation is 5.

Question: Did this participant score above or below the mean, and by how many standard deviations?

Solution

A Z-score of -1.5 is below the mean. In raw score terms: \(X = \mu + z\sigma = 40 + (-1.5)(5) = 40 - 7.5 = 32.5.\) So the participant scored 32.5, which is 1.5 standard deviations below the mean.

8. Z-Score Calculation

Z-Score with Negative Value

The mean stress score of a group is 25, with a standard deviation of 4. If a participant has a stress score of 18…

Question: Calculate their Z-score and interpret its meaning in the context of stress levels.

\(z = \frac{18 - 25}{4} = \frac{-7}{4} = -1.75.\) The Z-score is \(z \approx -1.75\), meaning this participant’s stress score is 1.75 standard deviations below the group mean — they have substantially lower stress than average. 

(18-25)/4
[1] -1.75

9. Z-Score Calculation

Multiple Z-Scores

In a clinical assessment, the average depression score is 60, with a standard deviation of 8. Two participants have scores of 72 and 48.

Question: Calculate the Z-scores for both participants and explain what the values tell us about their relative positions in the group.

  • Participant 1 (72): \(z_1 = \frac{72 - 60}{8} = 1.5\). Participant 2 (48): \(z_2 = \frac{48 - 60}{8} = -1.5\).
  • The first is 1.5 SD above the mean (higher depression than average), the second 1.5 SD below (lower). They are symmetric around the mean.
(72-60)/8;(48-60)/8
[1] 1.5
[1] -1.5

10. Z-Scores to Raw Scores

Working Backwards

The distribution of scores on a math test is approximately normal with mean \(\mu = 100\) and standard deviation \(\sigma = 15\).

Question: What raw score corresponds to \(z = 1.2\)?

Solve \(z = \frac{x-\mu}{\sigma}\) for \(x\): \(x = \mu + z\sigma = 100 + 1.2(15) = 118\) 

1.2*15 + 100
[1] 118

Reading the Z-Table

The table only ever gives the area to the left of z

  1. Sketch the curve and shade what the question asks for.
  2. Left of z: read the table value directly.
  3. Right of z: 1 minus the table value (subtract the area, not \(z\)).
  4. Between two z’s: big area minus small area.

Anxiety scores are approximately normal with \(\mu = 50\) and \(\sigma = 10\).

11. Probability: Left Tail

Below a Score

Question: What proportion of people score below 42?

  • \(z = \frac{42-50}{10} = -0.80\)
  • Shade the left tail. The table gives the left area directly: \(.2119\)
  • About 21.2% score below 42. 
pnorm(-0.80)                       # z-table
[1] 0.2118554
pnorm(42, mean = 50, sd = 10)      # unrounded z
[1] 0.2118554

12. Probability: Right Tail

Above a Score

Question: What proportion of people score above 65?

  • \(z = \frac{65-50}{10} = 1.50\)
  • Shade the right tail. The table gives the area to the left (\(.9332\)), so subtract from 1: \(1 - .9332 = .0668\)
  • About 6.7% score above 65. 
1 - pnorm(1.50)                                       # z-table
[1] 0.0668072
pnorm(65, mean = 50, sd = 10, lower.tail = FALSE)     # unrounded z
[1] 0.0668072

13. Probability: Between Two Scores

Between Two Values

Question: What proportion of people score between 45 and 60?

  • \(z_{45} = \frac{45-50}{10} = -0.50\) and \(z_{60} = \frac{60-50}{10} = 1.00\)
  • Table: left of \(1.00\) is \(.8413\); left of \(-0.50\) is \(.3085\)
  • Big area minus small area: \(.8413 - .3085 = .5328\)
  • About 53.3% score between 45 and 60. 
pnorm(1.00) - pnorm(-0.50)                            # z-table
[1] 0.5328072
pnorm(60, 50, 10) - pnorm(45, 50, 10)                 # unrounded z
[1] 0.5328072