📝 Chapter 6: The Normal Distribution

For test scores \( N(\mu = 75, \sigma = 10) \), find the z-score for \( x = 88 \).

From the standard normal table, \( P(Z < 2.0) \) ≈ ? (decimal to 4 places)

By the empirical rule, what percentage of values in a normal distribution fall within 2 standard deviations of the mean?

For \( X \sim N(50, 8) \), find \( P(X < 50) \).

A z-score of \( z = 1.5 \) places a student at what percentile? (round to nearest whole percent)

For \( X \sim N(100, 20) \), find \( P(80 < X < 120) \).

For \( X \sim \mathrm{Binomial}(50, 0.4) \), what are \( \mu \) and \( \sigma \) for the normal approximation?

Heights of adult women are normally distributed with \( \mu = 165 \) cm, \( \sigma = 6 \) cm. a) Find the percentile of a woman who is 175 cm tall. b) Find the height that is the 90th percentile (the height below which 90% of women fall).

A factory produces light bulbs whose lifetime is normally distributed with \( \mu = 1000 \) hours and \( \sigma = 100 \) hours. The company guarantees a refund if a bulb fails before 850 hours. What percentage of bulbs will require a refund?

A coin is flipped 200 times. Use the normal approximation to estimate \( P(X \le 90) \), where \( X \) is the number of heads. Apply the continuity correction.

For \( N(\mu, \sigma) \), determine the value \( a \) such that \( P(\mu - a\sigma < X < \mu + a\sigma) = 0.50 \). (Hint: \( P(Z < -a) = 0.25 \) so \( a \approx ? \).)

Explain in plain language what a z-score means. Why does standardizing make different normal distributions comparable? Use the example of test scores in two different classes.

A student claims: "If 95% of values are within 2 standard deviations, then 5% are MORE than 2 standard deviations from the mean — half on the left, half on the right, so 2.5% on each side." Is this correct? Explain why or why not, including the role of symmetry.

Describe a step-by-step procedure for using the normal approximation to the binomial. Include: (a) when the approximation is valid, (b) computing \( \mu \) and \( \sigma \), (c) the continuity correction. Apply your method to estimate \( P(X \ge 35) \) when \( X \sim \mathrm{Binomial}(50, 0.6) \).

Compare a discrete probability distribution and a continuous normal distribution. Why is \( P(X = a) = 0 \) for a continuous random variable, but not for a discrete one?

A package of cereal is supposed to weigh 500 g but the actual weight is normally distributed with \( \mu = 502 \) g, \( \sigma = 3 \) g. The company wants to know:
a) What proportion of packages weigh less than 495 g?
b) The weight that 99% of packages weigh more than (i.e., the 1st percentile).

SAT scores are normally distributed with \( \mu = 1500 \), \( \sigma = 200 \). A university requires applicants to score in the top 10%. What is the minimum SAT score required for admission?

A pharmaceutical company tests a new drug. The recovery time without treatment is \( N(14, 4) \) days. With the drug, it is \( N(10, 3) \) days. A patient recovered in 8 days. In which group is this more "typical" (closer to the mean)? Use z-scores to compare.

An election poll surveys 1000 randomly chosen voters; 520 say they support Candidate A. Assuming the true proportion of supporters is 50%, use the normal approximation to estimate the probability of getting at least 520 supporters in a poll. Comment on whether the result is statistically surprising (use a 5% threshold).