Let's go step by step to solve this problem about Evelyn's 400 meter dash times, which follow a normal distribution.
### Step 1: Understand the Problem
Evelyn's finishing times are normally distributed with:
- Mean time ([tex]\(\mu\)[/tex]) = 73 seconds
- Standard deviation ([tex]\(\sigma\)[/tex]) = 1.5 seconds
We need to determine how many of her 48 practice trials are expected to have finishing times between 71 and 76 seconds.
### Step 2: Convert the Finishing Time Boundaries to Z-Scores
For a normal distribution, the Z-score represents the number of standard deviations a value is from the mean. The formula for calculating a Z-score is:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\(X\)[/tex] is the value from the dataset.
- [tex]\(\mu\)[/tex] is the mean.
- [tex]\(\sigma\)[/tex] is the standard deviation.
For the lower bound (71 seconds):
[tex]\[ Z_{\text{lower}} = \frac{71 - 73}{1.5} = -1.333 \][/tex]
For the upper bound (76 seconds):
[tex]\[ Z_{\text{upper}} = \frac{76 - 73}{1.5} = 2.0 \][/tex]
### Step 3: Calculate the Cumulative Distribution Function (CDF) Values
The CDF of the standard normal distribution gives the probability that a standard normal random variable is less than or equal to a given value [tex]\( z \)[/tex].
For the lower Z-score:
[tex]\[ \text{CDF}(Z_{\text{lower}}) = \text{CDF}(-1.333) = 0.0912 \][/tex]
For the upper Z-score:
[tex]\[ \text{CDF}(Z_{\text{upper}}) = \text{CDF}(2.0) = 0.9772 \][/tex]
### Step 4: Calculate the Probability of Finishing Between 71 and 76 Seconds
The probability that a single trial will result in a finishing time between 71 and 76 seconds is the difference between the two CDF values:
[tex]\[ \text{Probability between 71 and 76 seconds} = \text{CDF}(Z_{\text{upper}}) - \text{CDF}(Z_{\text{lower}}) \][/tex]
[tex]\[ \text{Probability between 71 and 76 seconds} = 0.9772 - 0.0912 = 0.886 \][/tex]
### Step 5: Determine the Expected Number of Trials Within the Bounds
To find the expected number of trials within the 71 to 76 seconds range out of 48 trials, we multiply the probability by the total number of trials:
[tex]\[ \text{Expected trials} = 0.886 \times 48 = 42.53 \][/tex]
### Step 6: Round to the Nearest Whole Number
The expected number of trials needs to be rounded to the nearest whole number:
[tex]\[ \text{Expected trials} \approx 43 \][/tex]
### Final Answer
Therefore, Evelyn is expected to have approximately 43 trials out of 48 with finishing times between 71 and 76 seconds.
