A survey among freshmen at a certain university revealed that the number of hours spent studying the week before final exams was normally distributed with mean 25 and standard deviation 15. A sample of 36 students was selected. What is the probability that the average time spent studying for the sample was between 27.6 and 30 hours studying?
z = (29.6-25)/(15/sqrt(36)
z = 1.84
z = (30-25)/(15/sqrt(36)
z = 2
To find the probability that the average time spent studying for the sample was between 27.6 and 30 hours studying, we need to use the Central Limit Theorem, which states that for a large enough sample size, the distribution of sample means will be approximately normally distributed, regardless of the shape of the population distribution.
First, we need to calculate the mean and standard deviation of the sample means. The mean of the sample means is equal to the population mean, which is 25 in this case. The standard deviation of the sample means, also known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is 15 and the sample size is 36.
Standard error = population standard deviation / square root of sample size
Standard error = 15 / sqrt(36) = 2.5
Next, we need to convert the given values of 27.6 and 30 hours studying to Z-scores. The Z-score represents how many standard deviations a value is away from the mean in a normal distribution. The formula for calculating the Z-score is:
Z = (X - mean) / standard error
For 27.6 hours studying:
Z1 = (27.6 - 25) / 2.5
For 30 hours studying:
Z2 = (30 - 25) / 2.5
Now, we can use a standard normal distribution table or a calculator to find the probability associated with these Z-scores.
Using a standard normal distribution table, we can find the probabilities associated with the Z-scores. Let's assume that Pandas, a Python library, is being used to calculate the probabilities.
import pandas as pd
# Define the Z-scores
z1 = (27.6 - 25) / 2.5
z2 = (30 - 25) / 2.5
# Calculate the probabilities
probability = pd.DataFrame({
'Z-score': [z1, z2],
'Probability': [float('{:.4f}'.format(pd.read_csv('https://raw.githubusercontent.com/WizardOfAus/normal-distribution-table/master/normal-distribution-table.csv').iloc[int(z1 * 10)]['0']), 4)),
1 - float('{:.4f}'.format(pd.read_csv('https://raw.githubusercontent.com/WizardOfAus/normal-distribution-table/master/normal-distribution-table.csv').iloc[int(z2 * 10)]['0']), 4))]
})
probability
To find the probability that the average time spent studying for the sample was between 27.6 and 30 hours studying, we need to use the Central Limit Theorem.
The Central Limit Theorem states that for a large enough sample size, the distribution of the sample means will approach a normal distribution regardless of the shape of the population distribution.
In this case, the sample size is 36 students, which is considered to be sufficiently large for the Central Limit Theorem to apply.
First, we need to calculate the standard deviation of the sample mean, also known as the standard error of the mean. The formula for the standard error of the mean is:
Standard Error of the Mean (SE) = Standard Deviation (σ) / √(Sample Size)
Given that the standard deviation is 15 and the sample size is 36, we can calculate the standard error of the mean as:
SE = 15 / √36 = 15 / 6 = 2.5
Next, we can standardize the values of 27.6 and 30 using the standard error of the mean to find the z-scores.
Z-score = (X - Mean) / Standard Error of the Mean
For 27.6:
Z-score = (27.6 - 25) / 2.5 = 2.6 / 2.5 = 1.04
For 30:
Z-score = (30 - 25) / 2.5 = 5 / 2.5 = 2
Now, we can use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores.
Using a standard normal distribution table, we can find that the probability corresponding to a z-score of 1.04 is approximately 0.8508, and the probability corresponding to a z-score of 2 is approximately 0.9772.
Therefore, the probability that the average time spent studying for the sample was between 27.6 and 30 hours studying is:
0.9772 - 0.8508 = 0.1264, or 12.64%.