To determine the range within which approximately 95% of the data will lie for a normally distributed data set, we can use the concept of the confidence interval for a normal distribution.
Given:
- The mean ([tex]\(\mu\)[/tex]) = 52
- The standard deviation ([tex]\(\sigma\)[/tex]) = 4
### Step-by-Step Solution:
1. Understanding the 95% Confidence Interval:
- A 95% confidence interval means that approximately 95% of the data will lie within a certain number of standard deviations from the mean. For a normal distribution, this range corresponds to approximately [tex]\(\pm 1.96\)[/tex] standard deviations from the mean.
2. Calculating the Lower Bound:
- The lower bound can be found by subtracting [tex]\(1.96 \times \sigma\)[/tex] from the mean:
- [tex]\[ \text{Lower Bound} = \mu - (z \times \sigma) \][/tex]
- Here, [tex]\(z = 1.96\)[/tex], [tex]\(\mu = 52\)[/tex], and [tex]\(\sigma = 4\)[/tex]:
- [tex]\[ \text{Lower Bound} = 52 - (1.96 \times 4) \][/tex]
- [tex]\[ \text{Lower Bound} = 52 - 7.84 \][/tex]
- [tex]\[ \text{Lower Bound} = 44.16 \][/tex]
3. Calculating the Upper Bound:
- The upper bound can be found by adding [tex]\(1.96 \times \sigma\)[/tex] to the mean:
- [tex]\[ \text{Upper Bound} = \mu + (z \times \sigma) \][/tex]
- Here, [tex]\(z = 1.96\)[/tex], [tex]\(\mu = 52\)[/tex], and [tex]\(\sigma = 4\)[/tex]:
- [tex]\[ \text{Upper Bound} = 52 + (1.96 \times 4) \][/tex]
- [tex]\[ \text{Upper Bound} = 52 + 7.84 \][/tex]
- [tex]\[ \text{Upper Bound} = 59.84 \][/tex]
### Conclusion:
Approximately 95% of all cases for this normally distributed data will lie between the measures 44.16 and 59.84.
