Answer :
To determine the percentage of observations that lie between 0.372 and 0.428 for a normal distribution with a mean of 0.40 and a standard deviation of 0.028, follow these steps:
1. Understand the Problem:
- We have a normal distribution with:
- Mean (average) = 0.40
- Standard deviation = 0.028
- We need to find the percentage of data that falls between 0.372 and 0.428.
2. Calculate Z-scores:
- A z-score tells us how many standard deviations a data point is from the mean.
- Calculate the z-score for the lower bound (0.372):
[tex]\[
z_{\text{lower}} = \frac{0.372 - 0.40}{0.028}
\][/tex]
- Calculate the z-score for the upper bound (0.428):
[tex]\[
z_{\text{upper}} = \frac{0.428 - 0.40}{0.028}
\][/tex]
- The calculated z-scores are approximately -1.00 for the lower bound and 1.00 for the upper bound.
3. Find the Probability:
- We use the standard normal distribution table (or a statistical software tool) to find the probability associated with these z-scores.
- The probability of a z-score being less than 1.00 is about 0.8413.
- The probability of a z-score being less than -1.00 is about 0.1587.
4. Calculate the Percentage of Observations:
- Subtract the probability of the lower z-score from the probability of the upper z-score:
[tex]\[
\text{Probability} = 0.8413 - 0.1587 = 0.6826
\][/tex]
- Convert the probability to a percentage:
[tex]\[
0.6826 \times 100 = 68.26\%
\][/tex]
5. Conclusion:
- Approximately 68% of the observations will fall between 0.372 and 0.428.
- The correct answer choice is (a) 68%.
1. Understand the Problem:
- We have a normal distribution with:
- Mean (average) = 0.40
- Standard deviation = 0.028
- We need to find the percentage of data that falls between 0.372 and 0.428.
2. Calculate Z-scores:
- A z-score tells us how many standard deviations a data point is from the mean.
- Calculate the z-score for the lower bound (0.372):
[tex]\[
z_{\text{lower}} = \frac{0.372 - 0.40}{0.028}
\][/tex]
- Calculate the z-score for the upper bound (0.428):
[tex]\[
z_{\text{upper}} = \frac{0.428 - 0.40}{0.028}
\][/tex]
- The calculated z-scores are approximately -1.00 for the lower bound and 1.00 for the upper bound.
3. Find the Probability:
- We use the standard normal distribution table (or a statistical software tool) to find the probability associated with these z-scores.
- The probability of a z-score being less than 1.00 is about 0.8413.
- The probability of a z-score being less than -1.00 is about 0.1587.
4. Calculate the Percentage of Observations:
- Subtract the probability of the lower z-score from the probability of the upper z-score:
[tex]\[
\text{Probability} = 0.8413 - 0.1587 = 0.6826
\][/tex]
- Convert the probability to a percentage:
[tex]\[
0.6826 \times 100 = 68.26\%
\][/tex]
5. Conclusion:
- Approximately 68% of the observations will fall between 0.372 and 0.428.
- The correct answer choice is (a) 68%.
