The GPA at a particular school has an average of 2.89 with a standard deviation of 0.63. A random sample of 38 students from that school is collected. Find the probability that the average GPA for this sample is less than 2.8
To solve this problem, we will use the central limit theorem, which states that the distribution of sample means will be approximately normally distributed regardless of the shape of the population, as long as the sample size is sufficiently large (typically, at least 30) and random.
First, let's calculate the standard error of the mean (SEM), which is the standard deviation of the sampling distribution of the sample mean. The formula for SEM is given by:
SEM = standard deviation / square root of sample size
SEM = 0.63 / √38
SEM ≈ 0.102
Next, we need to standardize the sample mean using the formula:
Z = (sample mean - population mean) / SEM
Z = (2.8 - 2.89) / 0.102
Z ≈ -0.882
Finally, we can find the probability using a standard normal distribution table or calculator. The probability that the average GPA for this sample is less than 2.8 can be found by looking up the Z-value of -0.882 in the standard normal distribution table.
From the standard normal distribution table, we find that the area to the left of Z = -0.882 is approximately 0.1891.
Therefore, the probability that the average GPA for this sample is less than 2.8 is approximately 0.1891 or 18.91%.
To find the probability that the average GPA for the sample of 38 students is less than 2.8, we need to use the Central Limit Theorem.
The Central Limit Theorem states that for a large enough sample size (typically considered to be greater than 30), the distribution of sample means will approach a normal distribution, regardless of the shape of the population distribution.
1. Calculate the standard deviation of the sample mean:
The standard deviation of the sample mean, also known as the standard error, can be calculated using the formula:
Standard error = standard deviation / sqrt(sample size)
In this case, the standard deviation is 0.63 and the sample size is 38.
Standard error = 0.63 / sqrt(38) ≈ 0.1023
2. Calculate the z-score:
The z-score represents the number of standard deviations an individual sample mean is from the population mean. It can be calculated using the formula:
z = (sample mean - population mean) / standard error
In this case, the population mean is 2.89 and the desired sample mean is 2.8.
z = (2.8 - 2.89) / 0.1023 ≈ -0.878
3. Find the probability using a z-table:
Look up the z-score (-0.878) in the z-table to find the corresponding cumulative probability.
The cumulative probability represents the probability of getting a sample mean less than 2.8.
From the z-table, the corresponding cumulative probability for z = -0.878 is approximately 0.189. This means that the probability that the average GPA for the sample is less than 2.8 is 0.189, or 18.9%.
Therefore, the probability that the average GPA for the sample of 38 students is less than 2.8 is approximately 18.9%.
To find the probability that the average GPA for a sample is less than 2.8, we can use the central limit theorem and assume that the GPA follows a normal distribution.
First, we need to calculate the standard error of the mean (SEM), which represents the variability of sample means around the population mean. The formula for SEM is:
SEM = σ / sqrt(n)
Where:
σ is the standard deviation of the population (0.63 in this case)
sqrt represents the square root
n is the sample size (38 in this case)
Substituting the given values, we have:
SEM = 0.63 / sqrt(38)
≈ 0.102
Next, we need to standardize the sample mean using the z-score formula:
z = (x - μ) / SEM
Where:
x is the value we want to find the probability for (2.8 in this case)
μ is the population mean (2.89 in this case)
SEM is the standard error of the mean
Substituting the given values, we have:
z = (2.8 - 2.89) / 0.102
≈ -0.882
Now, we need to find the corresponding cumulative probability using a standard normal distribution table or a calculator. The cumulative probability represents the probability that a z-score is less than a given value.
Using the standard normal distribution table or a calculator, the cumulative probability for z = -0.882 is approximately 0.1889.
Therefore, the probability that the average GPA for this sample is less than 2.8 is approximately 0.1889, or 18.89%.