Scores on a test are normally distributed with a mean of 75 and a standard deviation of 8. Estimate the probability that a randomly selected student scored between 71 and 75.

http://davidmlane.com/hyperstat/z_table.html

0.19

To estimate the probability that a randomly selected student scored between 71 and 75, we need to find the area under the normal distribution curve between these two scores.

First, we need to standardize the scores by converting them to z-scores. A z-score represents the number of standard deviations a particular score is from the mean.

To find the z-score for 71, we use the formula:
z = (x - μ) / σ
where x is the score (71), μ is the mean (75), and σ is the standard deviation (8).

z = (71 - 75) / 8 = -0.5

To find the z-score for 75, we use the same formula:
z = (x - μ) / σ
where x is the score (75), μ is the mean (75), and σ is the standard deviation (8).

z = (75 - 75) / 8 = 0

Now we can find the probability using a standard normal distribution table or calculator. The probability of a z-score between -0.5 and 0 represents the probability that a randomly selected student scored between 71 and 75.

Using the standard normal distribution table, we find that the probability for a z-score of -0.5 is 0.3085, and the probability for a z-score of 0 is 0.5000.

To find the probability between these two z-scores, we subtract the probability associated with the lower z-score from the probability associated with the higher z-score:

0.5000 - 0.3085 = 0.1915

Therefore, the estimated probability that a randomly selected student scored between 71 and 75 is approximately 0.1915 or 19.15%.

To estimate the probability that a randomly selected student scored between 71 and 75, we need to calculate the z-scores for both scores and then use a standard normal distribution table or calculator.

The formula for calculating the z-score is:

z = (x - μ) / σ

where:
- x is the individual score
- μ is the population mean
- σ is the population standard deviation

For the lower score of 71:
z1 = (71 - 75) / 8 = -0.5

For the upper score of 75:
z2 = (75 - 75) / 8 = 0

Once we have the z-scores, we can use a standard normal distribution table or calculator to find the corresponding probabilities. The probability of a score occurring between the two z-scores is the difference between the two probabilities.

Using a standard normal distribution table, we can find the probabilities associated with the z-scores. Looking up the z-scores in the table, we get:

For z1 = -0.5, the probability is 0.3085 (or 30.85%)
For z2 = 0, the probability is 0.5000 (or 50%)

To find the probability between the two scores (71 and 75), we subtract the probability for z1 from the probability for z2:

Probability (71 ≤ score ≤ 75) = 0.5000 - 0.3085 = 0.1915 (or 19.15%)

Therefore, the estimated probability that a randomly selected student scored between 71 and 75 is approximately 0.1915 or 19.15%.