2.5 Points
30% of the fifth grade students in a large school district read below grade level. The distribution of sample proportions of samples of 100 students from this population is normal with a mean of 0.30 and a standard deviation of 0.045. Suppose that you select a sample of 100 fifth grade students from this district and find that the proportion that reads below grade level in the sample is 0.36. What is the probability that a second sample would be selected with a proportion less than 0.36?
To find the probability that a second sample would be selected with a proportion less than 0.36, we need to use the standard normal distribution and calculate the z-score.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
x is the observed proportion (0.36)
μ is the mean proportion (0.30)
σ is the standard deviation (0.045)
To find the probability, we need to calculate the area under the standard normal curve to the left of the z-score. This can be done using a standard normal distribution table or using statistical software.
Let's calculate the z-score:
z = (0.36 - 0.30) / 0.045
z = 1.33
Next, we find the probability by looking up the z-score in the standard normal distribution table. From the table, we find that the probability associated with a z-score of 1.33 is approximately 0.908.
Therefore, the probability that a second sample would be selected with a proportion less than 0.36 is approximately 0.908 or 90.8%.