Hypoglycemia is the most common serious side effect of insulin. It occurs in approximately \( 16 \% \) of Type I diabetic Consider a sample of 100 Type I diabetics taking insulin. Round to 4 decimal places (a) \( P \) (more than 25 having hypoglycemia \( )= \) (b) \( P(17 \) to 32 patients having hypoglycemia \( )= \)

To find the probabilities regarding hypoglycemia in a sample of 100 Type I diabetics on insulin, we can use the normal approximation to the binomial distribution since \( n \) is large (100 in this case) and \( p \) is not extremely close to 0 or 1. Here, the probability \( p \) of a patient experiencing hypoglycemia is approximately \( 0.16 \). First, we need the mean (\( \mu \)) and standard deviation (\( \sigma \)) of our binomial distribution: \[ \mu = np = 100 \times 0.16 = 16 \] \[ \sigma = \sqrt{np(1-p)} = \sqrt{100 \times 0.16 \times 0.84} \approx \sqrt{13.44} \approx 3.67 \] Now you can utilize the properties of the normal distribution for calculations. (a) To find \( P(\text{more than 25 having hypoglycemia}) \): Convert to the z-score for 25: \[ z = \frac{X - \mu}{\sigma} = \frac{25 - 16}{3.67} \approx 2.45 \] Using the z-table or a calculator, \( P(Z > 2.45) \) corresponds to approximately \( 0.0072 \). Thus, rounding to four decimal places: \[ P(\text{more than 25 having hypoglycemia}) \approx 0.0072 \] (b) To find \( P(17 \text{ to } 32 \text{ patients having hypoglycemia}) \): Calculate \( P(X < 32) \) and \( P(X < 16) \): For \( X = 32 \): \[ z = \frac{32 - 16}{3.67} \approx 4.36 \] \( P(Z < 4.36) \) is nearly 1. For \( X = 17 \): \[ z = \frac{17 - 16}{3.67} \approx 0.27 \] Using the z-table, \( P(Z < 0.27) \approx 0.6064 \). Now, subtract \( P(X < 17) \) from \( P(X < 32) \): \[ P(17 \text{ to } 32) = P(X < 32) - P(X < 17) \approx 1 - 0.6064 = 0.3936 \] Thus, rounding to four decimal places: \[ P(17 \text{ to } 32 \text{ patients having hypoglycemia}) \approx 0.3936 \]