The percent of fat calories that a person consumes each day is normally distributed with a mean of 37 and a standard deviation of 10. Suppose that 16 individuals are randomly chosen. Let X = average percent of fat calories. (a) For the group of 16, find the probability that the average percent of fat calories consumed is more than forty. (Round your answer to four decimal places.) 0.382 b) Find the first quartile for the average percent of fat calories. (Round your answer to two decimal places.) 30.25 percent of fat calories

Final answer:

The probability that the average percent of fat calories is more than 40 for a group of 16 individuals is 0.1151 and the first quartile for the average percent of fat calories is 34.82.

Explanation:

The problem presented is one of statistics, specifically dealing with the normal distribution. When dealing with the normal distribution, there are a few key facts to know. The mean (μ) is the average value of the data, the standard deviation (σ) measures the spread of the data, and the Z-score measures how many standard deviations an element is from the mean. It appears that you're asked to find the probability that the average percent of fat calories is more than 40 for a group of 16 individuals and the value corresponding to the first quartile.

1. For (a), remember that when dealing with averages, the standard deviation gets divided by the square root of the number of observations. Therefore, the new standard deviation becomes 10/√16 = 2.5. The Z-score can then be calculated by subtracting the mean from the observed value and dividing by the new standard deviation. That is, Z = (40 - 37) / 2.5 = 1.2. You can then refer to a Z-table to find the probability associated with this Z-score. However, since the question is asking for more than 40, you need to subtract the resulting probability from 1. Rounded to four decimal places, the probability that the average percent of fat calories is more than 40 is 0.1151.
2. For (b), the first quartile (Q1) is the value corresponding to the 25th percentile. This corresponds to a Z-score of -0.674 (from the Z-table). You can then use the formula Q1 = μ + Zσ to obtain Q1 = 37 - 0.674*2.5 = 34.815. Therefore, the first quartile for the average percent of fat calories is 34.82 (after rounding to two decimal places).

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Final answer:

The probability that the group consumes more than 40% of fat calories is 0.8849 (or 88.49%). The first quartile for the average percent of fat calories is 34.26%.

Explanation:

This problem requires the use of normal distributions and the calculation of z-scores, which is a quantitative measure that describes a value's relationship to the mean of a group of values. Knowing the mean (μ) and standard deviation (σ), you can calculate the z-score.

In part a) the problem asks for the probability that a sample of 16 individuals has an average fat calories consumption over 40%. First, we calculate the z-score.

z = (X - μ) / (σ/ √ n ), where n is the sample size. So, z = (40 - 37) / (10 / √ 16) = 1.2. Using a standard normal distribution table, we find that the probability corresponding to this z-value is ~0.1151. However, since the question is asking for more than 40%, we subtract this from 1 (1 - 0.1151 = 0.8849). So, the chance that the average of this sample would have more than 40% fat calories is 0.8849, or 88.49%.

For part b), the first quartile (also known as the 25th percentile) is the point at which 25% of the data fall below it. With a standard normal distribution, this z-value is approximately -0.674. We plug this into our z-score formula to find: X = μ + Zσ = 37 -0.674(10) =34.26%. Therefore, the point where 25% of individual average fat calories fall below it is 34.26%.

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