Question

In a sample of 100 adults, 7 were identified as having some allergies. Check whether the sample size is big enough so that you expect at least 10 successes and at least 10 failures. (2 points) n· p=7; n· (1-p)=93. One of the numbers is not larger than 10 , so the success-failure condition is not met.

Answer 1

To check whether the sample size is big enough to expect at least 10 successes and at least 10 failures, we can use the sample size \( n \) and the proportion of successes \( p \).

In this case, we have:

- \( n = 100 \) (total number of adults in the sample)
- \( x = 7 \) (number of adults with allergies)

Now, let's determine \( p \) (the proportion of successes, i.e., adults with allergies):

\[
p = \frac{x}{n} = \frac{7}{100} = 0.07
\]

Next, we calculate the expected number of successes (\( n \cdot p \)) and failures (\( n \cdot (1 - p) \)):

1. Expected number of successes:
\[
n \cdot p = 100 \cdot 0.07 = 7
\]

2. Expected number of failures:
\[
n \cdot (1 - p) = 100 \cdot (1 - 0.07) = 100 \cdot 0.93 = 93
\]

Now we can check the success-failure condition:

- We expected 7 successes (adults with allergies), which is less than 10.
- We expected 93 failures (adults without allergies), which is greater than 10.

Since we have fewer than 10 expected successes, the success-failure condition is not met.

Thus, the conclusion is that the sample size is not big enough to meet the success-failure condition.