PROBLEM 1: A researcher in a psychological lab investigated gender differences. She wished to compare male and female ability to recognize and remember visual details. She used 17 participants (8 males and 9 females) who were initially unaware of the actual experiment. First, she placed each one of them alone in a room with various objects and asked them to wait. After 10 min., she asked each of the participants to complete a 30-question posttest relating to several details in the room.
# Create the data frame
data <- data.frame(
Participant = 1:17,
Gender = c("M","M","M","M","M","M","M","M","F","F","F","F","F","F","F","F", "F"),
Posttest_Score = c(7,19,8,10,7,15,6,13,14,11,18,23,17,20,14,24,22)
)
data## Participant Gender Posttest_Score
## 1 1 M 7
## 2 2 M 19
## 3 3 M 8
## 4 4 M 10
## 5 5 M 7
## 6 6 M 15
## 7 7 M 6
## 8 8 M 13
## 9 9 F 14
## 10 10 F 11
## 11 11 F 18
## 12 12 F 23
## 13 13 F 17
## 14 14 F 20
## 15 15 F 14
## 16 16 F 24
## 17 17 F 221. Hypotheses:
\(H_o:\) There is no correlation between gender and visual detail recognition.
\(H_a:\) There is a correlation between gender and visual detail recognition
2. Level of Significance:
Set α = 0.05.
3. Appropriate Test Statistics:
As stated earlier, we decided to analyze the relationship between the two variables. A correlation will provide the relative strength of the relationship between the two variables. Gender is a discrete dichotomous variable and visual detail recognition is an interval scale variable. Therefore, we will use a point- biserial correlation.
5. Compute the test statistic:
# Data from Table 7.16
gender <- c(0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1, 1) # M = 0, F = 1
score <- c(7,19,8,10,7,15,6,13,14,11,18,23,17,20,14,24,22)
# Means
M_total <- mean(score)
M_male <- mean(score[gender == 0])
M_female <- mean(score[gender == 1])
# SD of scores
sd_total <- sd(score)
# Proportions
p_male <- mean(gender == 0)
p_female <- mean(gender == 1)
# Point-biserial correlation (manual formula)
r_pb <- (M_female - M_male) / sd_total * sqrt(p_male * p_female)
# Test statistic (t)
n <- length(score)
t_stat <- r_pb * sqrt((n - 2) / (1 - r_pb^2))
# Degrees of freedom
df <- n - 2
# Two-tailed p-value (manual)
p_value <- 2 * pt(-abs(t_stat), df)
# Print all
cat("Point-biserial correlation:", r_pb, "\n")## Point-biserial correlation: 0.6372441cat("Degrees of freedom:", df, "\n")## Degrees of freedom: 15cat("p-value (two-tailed):", p_value, "\n")## p-value (two-tailed): 0.0059335515. Critical Values or Rejection Region:
Table B.8 lists critical values for the Pearson product-moment correlation coefficient. Using the table of critical values requires that the degrees of freedom be known. Since df= n— 2 and n— 17, then df= 17— 2. Therefore, df= 15. Since we are conducting a two-tailed test and a = 0.05, the critical value is 0.482.
6. Statistical Decision:
Since \(r_{pb} = 0.637 > 0.482\), we must reject the null hypothesis.
7. Interpreting the Results:
We rejected the null hypothesis, suggesting that there is a significant and moderately strong correlation between gender and visual detail recognition.
8. Reporting the Results:
The results from the point-biserial correlation (\(r_{pb} = 0.637 > 0.482, p = 0.005933551 < 0.05\)) suggest that there is a strong relationship between gender and visual detail recognition. Moreover, the mean scores on the detail recognition test indicate that males (xM = 10.63) recalled fewer details, while females (xF = 18.11) recalled more details.