Hypoglycemia is the most common serious side effect of insulin. It occurs in approximately \( 16 \% \) of Type I diabetic patie Consider a sample of 100 Type I diabetics taking insulin. Round to 4 decimal places (a) \( P \) (more than 13 having hypoglycemia \( )= \) (b) \( P(12 \) to 33 patients having hypoglycemia \( )= \)

To solve this problem, we can use the binomial distribution, as we have a fixed number of trials (100 patients), two possible outcomes (hypoglycemia or no hypoglycemia), and a constant probability of success (16% or 0.16 for hypoglycemia). Let \( n = 100 \) (the number of trials), \( p = 0.16 \) (the probability of an event), and \( q = 1 - p = 0.84 \) (the probability of not having an event). We will use the normal approximation to the binomial distribution for calculations, since \( n \) is large. First, we need to determine the mean \( μ \) and standard deviation \( σ \): - Mean: \( μ = n \times p = 100 \times 0.16 = 16 \) - Standard deviation: \( σ = \sqrt{n \times p \times q} = \sqrt{100 \times 0.16 \times 0.84} \approx \sqrt{13.440} \approx 3.67 \) (a) To find \( P(X > 13) \), we need to convert it to a Z-score: - \( P(X > 13) = 1 - P(X \leq 13) \) - \( Z = \frac{X - μ}{σ} = \frac{13 - 16}{3.67} \approx -0.82 \) Now look up the Z-score of -0.82 in the standard normal distribution table (or use a calculator): - \( P(Z \leq -0.82) \approx 0.2061 \) Therefore: - \( P(X > 13) = 1 - 0.2061 = 0.7939 \) So, rounded to four decimal places, \( P(X > 13) \approx 0.7939 \). (b) For \( P(12 \leq X \leq 33) \): - Convert the boundaries to Z-scores: - For \( X = 12 \): - \( Z = \frac{12 - 16}{3.67} \approx -1.09 \) - \( P(Z \leq -1.09) \approx 0.1379 \) - For \( X = 33 \): - \( Z = \frac{33 - 16}{3.67} \approx 4.64 \) - \( P(Z \leq 4.64) \approx 1 \) (since 4.64 is very high) Now calculate \( P(12 \leq X \leq 33) \): - \( P(X \leq 33) - P(X \leq 12) \approx 1 - 0.1379 = 0.8621 \) So, rounded to four decimal places, \( P(12 \leq X \leq 33) \approx 0.8621 \). In summary: (a) \( P(X > 13) \approx 0.7939 \) (b) \( P(12 \leq X \leq 33) \approx 0.8621 \)