Answer :
To determine the percentage of individuals who score between Pat and Chris on a spatial abilities test with a mean of 80 and a standard deviation of 8, follow these steps:
1. Calculate the Z-scores:
The Z-score is calculated using the formula:
[tex]\[ Z = \frac{(X - \text{mean})}{\text{SD}} \][/tex]
Where:
- \( X \) is the individual score
- \(\text{mean}\) is the average score
- \(\text{SD}\) is the standard deviation
For Pat:
[tex]\[ Z_{Pat} = \frac{(76 - 80)}{8} = -0.5 \][/tex]
For Chris:
[tex]\[ Z_{Chris} = \frac{(94 - 80)}{8} = 1.75 \][/tex]
2. Find the Percentiles:
Using standard normal distribution tables or a calculator, you can find the percentiles corresponding to these Z-scores.
- For \( Z_{Pat} = -0.5 \), the percentile is approximately \( 30.85\% \).
- For \( Z_{Chris} = 1.75 \), the percentile is approximately \( 95.99\% \).
3. Calculate the Percentage of Individuals Scoring Between Pat and Chris:
Subtract the lower percentile (Pat’s percentile) from the upper percentile (Chris’s percentile):
[tex]\[ \text{Percentage between} = 95.99\% - 30.85\% = 65.14\% \][/tex]
4. Round to the Nearest Whole Number:
[tex]\[ \text{Rounded Percentage between} = 65\% \][/tex]
Therefore, 65% of individuals would score between Pat and Chris on this spatial abilities test. The correct answer is [tex]\( \boxed{65\%} \)[/tex].
1. Calculate the Z-scores:
The Z-score is calculated using the formula:
[tex]\[ Z = \frac{(X - \text{mean})}{\text{SD}} \][/tex]
Where:
- \( X \) is the individual score
- \(\text{mean}\) is the average score
- \(\text{SD}\) is the standard deviation
For Pat:
[tex]\[ Z_{Pat} = \frac{(76 - 80)}{8} = -0.5 \][/tex]
For Chris:
[tex]\[ Z_{Chris} = \frac{(94 - 80)}{8} = 1.75 \][/tex]
2. Find the Percentiles:
Using standard normal distribution tables or a calculator, you can find the percentiles corresponding to these Z-scores.
- For \( Z_{Pat} = -0.5 \), the percentile is approximately \( 30.85\% \).
- For \( Z_{Chris} = 1.75 \), the percentile is approximately \( 95.99\% \).
3. Calculate the Percentage of Individuals Scoring Between Pat and Chris:
Subtract the lower percentile (Pat’s percentile) from the upper percentile (Chris’s percentile):
[tex]\[ \text{Percentage between} = 95.99\% - 30.85\% = 65.14\% \][/tex]
4. Round to the Nearest Whole Number:
[tex]\[ \text{Rounded Percentage between} = 65\% \][/tex]
Therefore, 65% of individuals would score between Pat and Chris on this spatial abilities test. The correct answer is [tex]\( \boxed{65\%} \)[/tex].
