Question 12:
To determine the percentage of values that lie within the closed interval \([105, 135]\) for a normally distributed dataset with a mean of 120 and a standard deviation of 15, follow these steps:
1. Identify the parameters given:
- Population mean (\(\mu\)): 120
- Population standard deviation (\(\sigma\)): 15
- Lower bound: 105
- Upper bound: 135
2. Calculate the z-scores for the lower and upper bounds:
The z-score formula is given by:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where \(X\) is the value for which you want to find the z-score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
- Z-score for the lower bound (105):
[tex]\[
z_{105} = \frac{105 - 120}{15} = \frac{-15}{15} = -1
\][/tex]
- Z-score for the upper bound (135):
[tex]\[
z_{135} = \frac{135 - 120}{15} = \frac{15}{15} = 1
\][/tex]
3. Determine the cumulative probabilities corresponding to the z-scores:
Using the standard normal distribution table:
- The cumulative probability for \(z = -1\) is approximately 0.1587.
- The cumulative probability for \(z = 1\) is approximately 0.8413.
4. Calculate the probability of being between the bounds:
Subtract the cumulative probability of the lower bound from that of the upper bound:
[tex]\[
P(105 \leq X \leq 135) = P(Z \leq 1) - P(Z \leq -1) = 0.8413 - 0.1587 = 0.6826
\][/tex]
5. Convert the probability to a percentage:
[tex]\[
\text{Percentage} = 0.6826 \times 100 = 68.268\%
\][/tex]
Therefore, approximately 68.27\% of the values are in the closed interval \([105, 135]\). The closest provided answer is 68.
Question 13:
To find the median age of female patients diagnosed with breast cancer in a specific year, follow these steps:
1. Collect the age data:
Gather the ages of all female patients diagnosed with breast cancer over the specified period. Let's assume this data is collected accurately in a database or record system.
2. Arrange the ages in ascending order:
Sort the ages from the lowest to the highest.
3. Determine the median:
- If the number of ages is odd: The median is the middle value in the list.
- If the number of ages is even: The median is the average of the two middle values.
For instance, if the ages sorted are [30, 35, 40, 45, 50]:
- Since there are 5 values (odd number), the median is the middle value:
[tex]\[
\text{Median} = 40
\][/tex]
If the ages sorted are [30, 35, 40, 45, 50, 55]:
- Since there are 6 values (even number), the median is the average of the third and fourth values:
[tex]\[
\text{Median} = \frac{40 + 45}{2} = 42.5
\][/tex]
Thus, the median age provides a central value around which half the patients are younger, and half are older.
This is an overview of steps the clinic can follow to find the median age of the female patients diagnosed with breast cancer.
