the mean weight of 500 males students at a certain college is 151 lb and standard deviation is 15lb .assuming the weights are normally distributed, find how many students weigh between 120 and 155 lb ..ii) more than 185 lb

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

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To find the number of students who weigh between 120 and 155 pounds, we need to calculate the z-scores corresponding to these weights and then look up the corresponding probabilities from the standard normal distribution table.

Step 1: Calculate the z-score for 120 pounds.
The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the weight, μ is the mean weight, and σ is the standard deviation.

For 120 pounds:
z = (120 - 151) / 15
z = -31 / 15
z ≈ -2.07

Step 2: Look up the probability for the z-score -2.07 in the standard normal distribution table. The table will give us the probability of finding a value less than -2.07, so we need to subtract it from 0.5 to get the probability of finding a value between -2.07 and +2.07.

From the table, the probability for -2.07 is approximately 0.0192.

Step 3: Calculate the z-score for 155 pounds.
For 155 pounds:
z = (155 - 151) / 15
z = 4 / 15
z ≈ 0.27

Step 4: Look up the probability for the z-score 0.27 in the standard normal distribution table. This probability gives the percentage of values less than 0.27.

From the table, the probability for 0.27 is approximately 0.6064.

Step 5: Subtract the probability of finding a weight less than 120 pounds from the probability of finding a weight less than 155 pounds to get the probability of finding a weight between 120 and 155 pounds.

P(120 ≤ x ≤ 155) = P(x ≤ 155) - P(x ≤ 120)
P(120 ≤ x ≤ 155) = 0.6064 - 0.0192
P(120 ≤ x ≤ 155) ≈ 0.5872

Step 6: Multiply the probability by the total number of students to find the number of students who weigh between 120 and 155 pounds.

Number of students = Probability * Total number of students
Number of students = 0.5872 * 500
Number of students ≈ 293.6

Therefore, approximately 293 students weigh between 120 and 155 pounds.

ii) To find the number of students who weigh more than 185 pounds, we need to calculate the probability of finding a weight greater than 185 pounds and then multiply it by the total number of students.

Step 1: Calculate the z-score for 185 pounds.
For 185 pounds:
z = (185 - 151) / 15
z = 34 / 15
z ≈ 2.27

Step 2: Look up the probability for the z-score 2.27 in the standard normal distribution table. This probability gives the percentage of values less than 2.27, so we need to subtract it from 0.5 to get the probability of finding a value greater than 2.27.

From the table, the probability for 2.27 is approximately 0.9884.

Step 3: Calculate the number of students who weigh more than 185 pounds.

Number of students = (1 - Probability) * Total number of students
Number of students = (1 - 0.9884) * 500
Number of students ≈ 7.8

Therefore, approximately 8 students weigh more than 185 pounds.