Answer:
a. The distribution of X is a normal distribution (X-N) because the amount of coffee that people drink per day is normally distributed with a mean of 16 ounces and a standard deviation of 6.5 ounces. In a normal distribution, the values cluster around the mean and follow a bell-shaped curve.
b. The distribution of the sample mean, denoted by ?, is a normal distribution (?-NG). The sample mean is calculated by taking the average of the values in the sample. In this case, we are considering the average coffee consumption of the 32 randomly selected people.
c. To find the probability that one randomly selected person drinks between 15.5 and 16.1 ounces of coffee per day, we can use the standard normal distribution table or a statistical calculator. First, we need to standardize the values using the formula:
Z = (X - ?) / σ
where X is the value we want to find the probability for, ? is the mean, and σ is the standard deviation.
Using this formula, we calculate the z-scores for 15.5 and 16.1:
Z1 = (15.5 - 16) / 6.5
Z2 = (16.1 - 16) / 6.5
Next, we can look up the corresponding probabilities in the standard normal distribution table for these z-scores. The probability that one randomly selected person drinks between 15.5 and 16.1 ounces of coffee per day is equal to the difference between these two probabilities.
d. To find the probability that the average coffee consumption for the 32 people is between 15.5 and 16.1 ounces per day, we use the central limit theorem. The central limit theorem states that for a large enough sample size, the distribution of the sample mean approximates a normal distribution, regardless of the shape of the population distribution.
In this case, since we have a sample size of 32, we can use the normal distribution to approximate the distribution of the sample mean. We calculate the z-scores for 15.5 and 16.1 using the formula:
Z1 = (15.5 - ?) / (σ / √n)
Z2 = (16.1 - ?) / (σ / √n)
where ? is the population mean, σ is the population standard deviation, and n is the sample size.
We can then find the corresponding probabilities in the standard normal distribution table for these z-scores and subtract them to find the probability that the average coffee consumption is between 15.5 and 16.1 ounces per day.
e. Yes, the assumption that the distribution is normal is necessary for part d). The central limit theorem assumes that the population distribution is approximately normal in order to apply the normal distribution approximation to the sample mean. If the population distribution is not normal, the results obtained using the normal distribution approximation may not be accurate.
f. To find the IQR (interquartile range) for the average of 32 coffee drinkers, we first need to calculate the quartiles. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half of the data.
Since we are dealing with the average coffee consumption, we need to find the quartiles of the sample mean. We can use the fact that the mean and standard deviation of the sample mean are equal to the population mean and population standard deviation divided by the square root of the sample size, respectively.
Q1 = ? - (1.349 * σ / √n)
Q3 = ? + (1.349 * σ / √n)
where ? is the population mean, σ is the population standard deviation, and n is the sample size.
The IQR is then calculated as Q3 - Q1.
Explanation: