The percent of fat calories that a person in America consumes each day is normally distributed with a mean of 36 and a standard deviation of about 10. Suppose that 16 individuals are randomly chosen for an experiment.
Completely describe the distribution of sample means for the group of 16 individuals that was chosen in the experiment above.
Find the probability that the percent of fat calories that an individual in the group consumes is more than 30 percent per day.
For the group of 16, find the probability that the average percent of fat calories consumed is more than 30 calories.
Find the first quartile for the average percent of fat calories consumed for the group of 16.
To describe the distribution of sample means for the group of 16 individuals, we need to calculate the mean and standard deviation for the sample mean.
The mean of the sample means would be equal to the mean of the population, which is 36.
To find the standard deviation of the sample mean, also known as the standard error, we divide the standard deviation of the population by the square root of the sample size. In this case, the standard deviation of the population is 10, and the sample size is 16. Therefore, the standard error is 10 / √16 = 10 / 4 = 2.5.
So, the distribution of sample means for the group of 16 individuals has a mean of 36 and a standard deviation of 2.5.
To find the probability that an individual in the group consumes more than 30 percent of fat calories per day, we need to convert this value to a z-score and then find the corresponding probability from the standard normal distribution.
The formula to calculate the z-score is:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean of the population, and σ is the standard deviation of the population.
Plugging in the values, we get:
z = (30 - 36) / 10 = -6 / 10 = -0.6
We can use a z-table or a calculator to find the probability corresponding to a z-score of -0.6. Using a standard normal distribution table, we can find that the probability is approximately 0.2743.
So, the probability that an individual in the group consumes more than 30 percent of fat calories per day is approximately 0.2743.
To find the probability that the average percent of fat calories consumed is more than 30 percent for the group of 16, we use the same approach, but this time we use the distribution of sample means.
The formula to calculate the z-score for the sample mean is:
z = (x - μ) / σ
where x is the value we want to find the probability for (30 in this case), μ is the mean of the sample means (36), and σ is the standard error (2.5).
Plugging in the values, we get:
z = (30 - 36) / 2.5 = -6 / 2.5 = -2.4
Using a standard normal distribution table or a calculator, we can find that the probability corresponding to a z-score of -2.4 is approximately 0.0082.
So, the probability that the average percent of fat calories consumed is more than 30 percent for the group of 16 is approximately 0.0082.
To find the first quartile for the average percent of fat calories consumed for the group of 16, we need to find the value below which 25% of the sample means fall.
Using the z-score table, we can find that the z-score corresponding to the first quartile (25th percentile) is approximately -0.674.
The formula to calculate the actual value of the first quartile is:
x = μ + z * σ
where μ is the mean of the sample means (36), z is the z-score (-0.674), and σ is the standard error (2.5).
Plugging in the values, we get:
x = 36 + (-0.674) * 2.5 = 36 - 1.685 = 34.315
So, the first quartile for the average percent of fat calories consumed for the group of 16 is approximately 34.315.
The distribution of sample means for the group of 16 individuals can be described as follows:
1. Mean: The mean of the sample means will be equal to the population mean, which is 36.
2. Standard Deviation: The standard deviation of the sample means, also known as the standard error, can be calculated by dividing the population standard deviation (10) by the square root of the sample size (16): 10 / √16 = 10 / 4 = 2.5.
3. Shape: Since the sample size is large (n > 30), the distribution of sample means can be approximated to a normal distribution, according to the Central Limit Theorem.
Next, let's find the probability that an individual in the group consumes more than 30% of fat calories per day:
1. Z-score: To calculate the probability, we need to convert the value of 30% into a Z-score. The Z-score formula is given by: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. Substituting the values, we get Z = (30 - 36) / 10 = -0.6.
2. Probability calculation: We can use a Z-table or calculator to find the probability associated with a Z-score of -0.6, which would give us the probability that an individual in the group consumes less than 30% of fat calories per day. Subtracting this probability from 1 will give us the probability that an individual consumes more than 30%. Let's assume this probability is P(X > 30).
Next, let's find the probability that the average percent of fat calories consumed for the group of 16 is more than 30%:
1. Central Limit Theorem: According to the Central Limit Theorem, the distribution of the sample means will be approximately normally distributed, so we can use the Z-score formula again.
2. Z-score calculation: Using the same formula as before, Z = (30 - 36) / (10 / √16) = -1.6.
3. Probability calculation: Like before, we can use a Z-table or calculator to find the probability associated with a Z-score of -1.6, which would give us the probability that the average percent of fat calories consumed for the group of 16 is less than 30%. Subtracting this probability from 1 will give us the probability that the average percent of fat calories consumed is more than 30%. Let's assume this probability is P(X̄ > 30).
Finally, let's find the first quartile for the average percent of fat calories consumed for the group of 16:
1. Z-score calculation: The first quartile corresponds to a cumulative probability of 0.25. Using a Z-table or calculator, we can find the Z-score associated with this cumulative probability, which would give us the Z-score for the first quartile.
2. Reverse Z-score formula: Using the Z-score formula, we can find the value for X, which corresponds to the first quartile. The formula is given by: X = μ + (Z * σ).
Note: Plug in the values of μ (36), σ (10 / √16), and the Z-score for the first quartile obtained from the Z-table or calculator, which will give us the first quartile for the average percent of fat calories consumed for the group of 16.