Let's go through this problem step by step.
1. First, identify the sample size [tex]\( n \)[/tex]:
[tex]\[
n = 100
\][/tex]
2. Next, we know the number of individuals with allergies, which we'll call the number of successes [tex]\( k \)[/tex]:
[tex]\[
k = 7
\][/tex]
3. Calculate the probability of success [tex]\( p \)[/tex], which is the proportion of individuals with allergies:
[tex]\[
p = \frac{k}{n} = \frac{7}{100} = 0.07
\][/tex]
4. Calculate the probability of failure [tex]\( q \)[/tex], which is the complement of [tex]\( p \)[/tex]:
[tex]\[
q = 1 - p = 1 - 0.07 = 0.93
\][/tex]
5. Determine the expected number of successes [tex]\( n \cdot p \)[/tex]:
[tex]\[
n \cdot p = 100 \cdot 0.07 = 7.000000000000001 \approx 7
\][/tex]
6. Determine the expected number of failures [tex]\( n \cdot (1 - p) \)[/tex]:
[tex]\[
n \cdot (1 - p) = 100 \cdot 0.93 = 93
\][/tex]
7. Check the success-failure condition:
- The expected number of successes should be at least 10:
[tex]\[
7 < 10
\][/tex]
- The expected number of failures should be at least 10:
[tex]\[
93 \geq 10
\][/tex]
We see that the expected number of successes is less than 10 while the expected number of failures is greater than 10.
Since the expected number of successes (7) is not larger than 10, the success-failure condition is not met.
So, the sample size is not big enough to expect at least 10 successes and 10 failures.
Final answers:
- [tex]\( n \cdot p = 7.000000000000001 \)[/tex]
- [tex]\( n \cdot (1-p) = 93.0 \)[/tex]
- One of the numbers is not larger than 10, so the success-failure condition is not met.
