Question

Let X equal the weight of the soap in a "6-pound" box. Assume that the distribution of X is N(6.05, 0.0004). a) Find P(X < 6.0171). b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. HINT: Let Y equal the number of boxes that weigh less than 6.0171 pounds. c) Let Xbar be the sample mean of the nine boxes. Find P(Xbar ≤ 6.035).

Answer 1

Use the binomial distribution to find the probability that at most two boxes weigh less than 6.0171 pounds. Use the central limit theorem to find P(x ≤ 6.035).

Step-by-step explanation:

The questions can be answered as -

a) To find P(X < 6.0171), we can use the standard normal distribution table or a calculator that can calculate normal probabilities. We subtract the mean from 6.0171 and divide by the standard deviation to get the z-score. Then we can look up the z-score in the standard normal distribution table to find the corresponding probability. Alternatively, using a calculator, we can use the cumulative distribution function with the mean, standard deviation, and 6.0171 to find the probability. The result is the probability that X is less than 6.0171.

b) To find the probability that at most two boxes weigh less than 6.0171 pounds each, we can use the binomial distribution. Let Y be the number of boxes that weigh less than 6.0171 pounds. We want to find P(Y ≤ 2). We can use the binomial probability formula to calculate the probability of each possible value of Y (0, 1, and 2) and sum them up.

c) To find P(x ≤ 6.035), we can use the central limit theorem. The sample mean of the nine boxes, x, follows a normal distribution with mean 6.05 and standard deviation 0.0004/sqrt(9). We can use the standard normal distribution table or a calculator to find the probability that x is less than or equal to 6.035.