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%.
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
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