Answer:
Therefore, the reading score below which 75% of the reading scores fall is approximately 24.02.
Explanation:
a) Find the z-score for a reading score of 25.3.
To find the z-score, we use the formula:
z = (x - μ) / σ
where z is the z-score, x is the reading score, μ is the mean, and σ is the standard deviation.
Plugging in the values, we have:
z = (25.3 - 20.7) / 5.8
Calculating this, we get:
z ≈ 0.7931
Therefore, the z-score for a reading score of 25.3 is approximately 0.7931.
(b) Find the reading score that corresponds to a z-score of -1.5.
To find the reading score, we rearrange the formula:
x = μ + z * σ
Plugging in the values, we have:
x = 20.7 + (-1.5) * 5.8
Calculating this, we get:
x ≈ 11.2
Therefore, the reading score that corresponds to a z-score of -1.5 is approximately 11.2.
(c) What proportion of reading scores is between 11.2 and 25.3?
To find the proportion, we need to find the area under the normal distribution curve between these two scores. We can use a standard normal distribution table or a calculator to find the z-scores and then subtract the corresponding proportions.
Using a calculator, we find the z-scores:
z1 ≈ -1.5
z2 ≈ 0.7931
Then, we can use the calculator or table to find the proportions:
P(11.2 < x < 25.3) = P(z1 < z < z2)
Calculating this, we get:
P(11.2 < x < 25.3) ≈ 0.7603
Therefore, approximately 76.03% of reading scores are between 11.2 and 25.3.
(d) Find the reading score below which 75% of the reading scores fall.
To find the reading score, we can use the cumulative distribution function (CDF) of the standard normal distribution. We need to find the z-score that corresponds to the 75th percentile.
Using a calculator or table, we find the z-score:
z ≈ 0.6745
Then, we can use the formula to find the reading score:
x = μ + z * σ
Plugging in the values, we have:
x = 20.7 + 0.6745 * 5.8
Calculating this, we get:
x ≈ 24.02