To find the probability that the number of successes [tex]\( X \)[/tex] is between 30 and 40 using the normal approximation, we'll follow these steps:
### Step 1: Calculate the Mean and Standard Deviation
For a binomial distribution, the mean [tex]\( \mu \)[/tex] and standard deviation [tex]\( \sigma \)[/tex] are given by:
[tex]\[
\mu = n \cdot p
\][/tex]
[tex]\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\][/tex]
Given:
[tex]\[
n = 78, \quad p = 0.43
\][/tex]
we find:
[tex]\[
\mu = 78 \cdot 0.43 = 33.54
\][/tex]
[tex]\[
\sigma = \sqrt{78 \cdot 0.43 \cdot (1 - 0.43)} = \sqrt{78 \cdot 0.43 \cdot 0.57} \approx 4.3724
\][/tex]
### Step 2: Standardize the Bounds
To find the probability [tex]\( P(30 < X < 40) \)[/tex], we need to standardize these bounds by converting them to z-scores using the following formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
For the lower bound, [tex]\( X = 30 \)[/tex]:
[tex]\[
z_{\text{lower}} = \frac{30 - 33.54}{4.3724} \approx -0.8096
\][/tex]
For the upper bound, [tex]\( X = 40 \)[/tex]:
[tex]\[
z_{\text{upper}} = \frac{40 - 33.54}{4.3724} \approx 1.4775
\][/tex]
### Step 3: Find the Cumulative Probabilities
Using a standard normal distribution table or a calculator, we can find the cumulative probabilities corresponding to these z-scores.
The cumulative probability [tex]\( P(Z < z_{\text{upper}}) \)[/tex] for [tex]\( z_{\text{upper}} \approx 1.4775 \)[/tex] is:
[tex]\[
P(Z < 1.4775) \approx 0.9306
\][/tex]
The cumulative probability [tex]\( P(Z < z_{\text{lower}}) \)[/tex] for [tex]\( z_{\text{lower}} \approx -0.8096 \)[/tex] is:
[tex]\[
P(Z < -0.8096) \approx 0.2095
\][/tex]
### Step 4: Calculate the Probability
Finally, the probability that [tex]\( X \)[/tex] is between 30 and 40 is the difference between these cumulative probabilities:
[tex]\[
P(30 < X < 40) = P(Z < z_{\text{upper}}) - P(Z < z_{\text{lower}})
\][/tex]
[tex]\[
P(30 < X < 40) = 0.9306 - 0.2095 = 0.7211
\][/tex]
### Answer
The probability [tex]\( P(30 < X < 40) \)[/tex] is approximately [tex]\( 0.7211 \)[/tex].
