If a normally distributed group of test scores have a mean of 76 and a standard deviation of 10. Find the probability that student will earn a grade between 65 and 85

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To find the probability that a student will earn a grade between 65 and 85, we will use the properties of the normal distribution.

Step 1: Standardize the values
To standardize the values, we use the formula:
z = (x - μ) / σ
where z is the standard score, x is the value of interest, μ is the mean, and σ is the standard deviation.

For the lower bound:
z_lower = (65 - 76) / 10 = -1.1

For the upper bound:
z_upper = (85 - 76) / 10 = 0.9

Step 2: Look up the area under the standard normal curve
Using a standard normal table or a calculator, we can find the area under the curve between these two z-values.

The area to the left of z = -1.1 is 0.1357.
The area to the left of z = 0.9 is 0.8159.

Step 3: Calculate the probability between the two values
To find the probability between these two values, we subtract the area to the left of the lower bound from the area to the left of the upper bound.

P(65 ≤ X ≤ 85) = P(X ≤ 85) - P(X ≤ 65)
= 0.8159 - 0.1357
= 0.6802

Therefore, the probability that a student will earn a grade between 65 and 85 is 0.6802, or 68.02%.

To find the probability that a student will earn a grade between 65 and 85, given a normally distributed group of test scores with a mean of 76 and a standard deviation of 10, you can use the Z-score formula and the standard normal distribution table.

First, we need to calculate the Z-score for both 65 and 85. The Z-score indicates how many standard deviations an individual data point is from the mean.

The Z-score formula is:

Z = (X - μ) / σ

Where:
- Z is the Z-score
- X is the data point (in this case, 65 or 85)
- μ is the mean of the distribution (76 in this case)
- σ is the standard deviation of the distribution (10 in this case)

For 65:
Z = (65 - 76) / 10 = -1.1

For 85:
Z = (85 - 76) / 10 = 0.9

Now, we can use the standard normal distribution table to find the probability associated with these Z-scores. The table provides the cumulative probability up to a given Z-score.

Looking up the Z-score of -1.1 in the standard normal distribution table, the cumulative probability is approximately 0.1357.

Looking up the Z-score of 0.9, the cumulative probability is approximately 0.8159.

To find the probability between 65 and 85, we need to subtract the smaller cumulative probability from the larger cumulative probability.

Probability = 0.8159 - 0.1357 = 0.6802

So, the probability that a student will earn a grade between 65 and 85 is approximately 0.6802.