Question

The population of weights of a particular fruit is normally distributed, with a mean of 223 grams and a standard deviation of 14 grams. If 19 fruits are picked at random, then 4% of the time, their mean weight will be greater than how many grams? Round your answer to the nearest gram.

Answer 1

The weight threshold exceeded only 4% of the time by the mean weight of 19 randomly selected fruits is calculated using the z-score corresponding to the 96th percentile and the standard error of the mean.

Step-by-step explanation:

To find the threshold weight such that 4% of the time, the mean weight of 19 randomly picked fruits will be greater than this weight, we need to use the concept of the sampling distribution of the sample mean.

First, we calculate the standard error of the mean which is the standard deviation divided by the square root of the sample size (n = 19): standard error = 14 / √19. Then, we look for the z-score that corresponds to the 96th percentile (since 4% of the values lie above it) of the standard normal distribution. Using a z-table, we find that this z-score is approximately 1.75.

Now we can calculate the threshold weight: threshold weight = mean + z * standard error, which gives us the mean weight that is exceeded only 4% of the time for the sample of 19 fruits.

Rounding to the nearest gram, we will use the calculated standard error and the z-score to determine the exact weight.