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the scores obtained by 800 students in statistics exam are normally distributed with a mean of 78 and a standard deviation of 7 find the probability value and determine the number of students who obtained a A.) 80 and above B.) 75 and below C.) from 71 to 85​

Sagot :

Mags67idn

Answer:

A.) Probability of scoring 80 and above = 0.3881, Number of students = 310                                                                                                                      B.) Probability of scoring 75 and below = 0.3336, Number of students = 268                                                                                                                  C.) Probability of scoring from 71 to 85 = 0.6826, Number of students = 546

Step-by-step explanation:

Given: Mean (μ) = 78                                                                                     Standard deviation (σ) = 7                                                                               Total number of students = 800

A.) To find the probability of students scoring 80 and above: First, we need to find the Z-score for a score of 80:

Z = X-μ/σ = 80-78/7 = 2/7 = 0.2857

Now, we find the probability using a Z-table: ( > 0.2857) = 1 − ( < 0.2857)

P ( Z > 0.2857) = 1− P(Z<0.2857) (>0.2857) ≈ 1 −0.6119 ≈ 0.3881

To find the number of students who scored 80 and above:

= ∗

= 800 ∗ 0.3881 ≈ 310.48 ≈ 310

B.) To find the probability of students scoring 75 and below: Using the Z-score formula:

Z = X- μ/σ = 75-78/7 = -0.4286

Now, we find the probability using a Z-table:

P ( Z < −0.4286) ≈ 0.3336

To find the number of students who scored 75 and below:

= ∗

= 800 ∗ 0.3336 ≈ 267.68 ≈ 268

C.) To find the probability of students scoring from 71 to 85: We need to find the Z-scores for 71 and 85:

For 71:  Z = 71-78/7 = -1

For 85: Z = 85-78/7 = 1

Now, find the probability of Z being between -1 and 1 using the Z-table:

P ( −1 < Z <1 ) ≈ P ( Z < 1 ) − P ( Z < −1 ) = 0.8413 −0.1587 =0.6826

To find the number of students who scored from 71 to 85:

= ∗

= 800 ∗ 0.6826 ≈ 546.08 ≈ 546