the scores of a recent test taken by 500 students had an approximately normal distribution with a mean of 75…

the scores of a recent test taken by 500 students had an approximately normal distribution with a mean of 75 and a standard deviation of 4.6. determine the number of students who scored between 70 and 85 (use desmos - round to the whole number)

the scores of a recent test taken by 500 students had an approximately normal distribution with a mean of 75 and a standard deviation of 4.6. determine the number of students who scored between 70 and 85 (use desmos - round to the whole number)

Answer

Explanation:

Step1: Calculate z - scores

For (x = 70), the z - score (z_1=\frac{70 - 75}{4.6}=\frac{- 5}{4.6}\approx - 1.09). For (x = 85), the z - score (z_2=\frac{85 - 75}{4.6}=\frac{10}{4.6}\approx2.17).

Step2: Find the probabilities

Using the standard normal distribution table or a calculator (like in Desmos), (P(Z < - 1.09)\approx0.1379) and (P(Z < 2.17)\approx0.9850). The probability (P(-1.09<Z<2.17)=P(Z < 2.17)-P(Z < - 1.09)=0.9850 - 0.1379 = 0.8471).

Step3: Calculate the number of students

The total number of students is (n = 500). The number of students who scored between 70 and 85 is (n\times P(-1.09 < Z < 2.17)=500\times0.8471 = 423.55\approx424).

Answer:

424