The average grade point average(GPA)of undergraduate students in New York is normally distributed with a population mean of and a population standard deviation of. Compute the following,
1)The percentage of students with GPA'S between 2.0 and 2.6
2)If a sample of 49 students is taken, what is the probability that the sample mean GPA will be between 2.60 and 2.70?
To compute the requested probabilities regarding the GPA of undergraduate students in New York, we need to use the properties of the normal distribution. We will assume that the population mean and standard deviation were accidentally omitted from the question. For the purpose of explanation, let's say the population mean is μ and the population standard deviation is σ.
1) To find the percentage of students with GPAs between 2.0 and 2.6, we need to calculate the area under the normal distribution curve between those two values. This can be done by calculating the z-scores corresponding to each GPA and then finding the difference in cumulative probabilities.
First, we need to standardize the GPAs by calculating the z-scores. The formula for calculating the z-score is:
z = (X - μ) / σ
Let's assume μ = 2.4 and σ = 0.2 for this example.
For GPA 2.0:
z1 = (2.0 - 2.4) / 0.2
z1 = -2
For GPA 2.6:
z2 = (2.6 - 2.4) / 0.2
z2 = 1
Now, we can use a standard normal distribution table or a calculator to look up the cumulative probabilities associated with these z-scores.
Once you find the cumulative probabilities corresponding to z1 and z2, subtract the cumulative probability associated with z1 from the cumulative probability associated with z2. The result will give you the percentage of students with GPAs between 2.0 and 2.6.
2) To find the probability that the sample mean GPA will be between 2.60 and 2.70, we need to calculate the standard error of the mean (SE) and then find the z-scores for these values.
The formula for the standard error of the mean is:
SE = σ / √n
In this case, n = 49 (the sample size). Let's assume σ = 0.2 for this example.
SE = 0.2 / √49
SE ≈ 0.0286
Next, we need to calculate the z-scores for 2.60 and 2.70 using the formula mentioned earlier:
For GPA 2.60:
z1 = (2.60 - μ) / SE
For GPA 2.70:
z2 = (2.70 - μ) / SE
Once you find the z-scores, you can use a standard normal distribution table or a calculator to find the cumulative probabilities associated with these z-scores.
Finally, subtract the cumulative probability associated with z1 from the cumulative probability associated with z2 to obtain the probability that the sample mean GPA will be between 2.60 and 2.70.
Please note that the exact values of μ and σ are required to calculate the probabilities accurately.