Question

course: probability and statistics
1-use a continuity correction to convert each binomial probability to a normal distribution probability
a)-the probability of getting between 280 and 320 successes, inclusive.
b-the probability of getting between 290 and 305 successes, inclusive
2-in a survey of 8- to 18-year-old heavy media users in the united states, 47% said they get fair or poor grades (c and below). you randomly select forty-five 8- to 18-year-old heavy media users in the united states and ask them whether they get fair or poor grades. what is the probability that fewer than 30 of them respond yes?

Explanation:

Problem 1a: Continuity Correction for Binomial to Normal

Step1: Recall Continuity Correction

For \( P(a \leq X \leq b) \) in binomial, when converting to normal, we use \( P(a - 0.5 < X < b + 0.5) \) (since inclusive, we adjust the endpoints by 0.5). Here, \( a = 280 \), \( b = 320 \).

Step2: Apply Correction

So the normal probability is \( P(279.5 < X < 320.5) \).

Step1: Recall Continuity Correction

For \( P(a \leq X \leq b) \) in binomial, adjust endpoints by 0.5. Here, \( a = 290 \), \( b = 305 \).

Step2: Apply Correction

The normal probability is \( P(289.5 < X < 305.5) \).

Step1: Identify Binomial Parameters

Let \( X \) be the number of successes (saying fair/poor grades). \( n = 45 \), \( p = 0.47 \). Check if normal approximation is appropriate: \( np = 45 \times 0.47 = 21.15 \), \( n(1 - p) = 45 \times 0.53 = 23.85 \), both \( \geq 5 \), so normal approximation is okay.

Step2: Find Mean and Standard Deviation

Mean \( \mu = np = 21.15 \), Standard deviation \( \sigma = \sqrt{np(1 - p)} = \sqrt{45 \times 0.47 \times 0.53} \approx \sqrt{21.15 \times 0.53} \approx \sqrt{11.2095} \approx 3.348 \).

Step3: Apply Continuity Correction

"Fewer than 30" means \( X < 30 \), so continuity correction gives \( X < 29.5 \).

Step4: Calculate Z - Score

\( Z = \frac{29.5 - \mu}{\sigma} = \frac{29.5 - 21.15}{3.348} = \frac{8.35}{3.348} \approx 2.49 \).

Step5: Find Probability from Z - Table

\( P(Z < 2.49) \approx 0.9936 \) (using Z - table: for \( Z = 2.49 \), the area to the left is approximately 0.9936).

Answer:

\( P(279.5 < X < 320.5) \)

Problem 1b: Continuity Correction for Binomial to Normal