The weights of the 100 students in an introductory statistics class are normally distributed, with a mean of 170 pounds and a standard deviation of 5 pounds.

How many students would you expect to have weight between 162 pounds and 178 pounds?

What is the probability that a student picked at random weighs less than 165 pounds?

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

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To answer the first question, we need to find the proportion of students with weights between 162 and 178 pounds.

To do this, we will use the formula for the standard normal distribution:
Z = (X - μ) / σ

Where Z is the z-score, X is the value we want to find the proportion for, μ is the mean, and σ is the standard deviation.

First, let's convert the values of 162 and 178 to z-scores:

Z1 = (162 - 170) / 5
Z2 = (178 - 170) / 5

Next, we need to find the cumulative proportion of students for each z-score using a standard normal distribution table or a calculator.

Using the z-score table, we can find that the cumulative proportion corresponding to Z1 is approximately 0.1587 and the cumulative proportion corresponding to Z2 is approximately 0.8413.

To find the proportion of students with weights between 162 and 178 pounds, we subtract the cumulative proportion for Z1 from the cumulative proportion for Z2:

Proportion = 0.8413 - 0.1587

Therefore, we would expect approximately 0.6826 (68.26%) of the 100 students to have weights between 162 and 178 pounds.

For the second question, we need to find the probability that a randomly picked student weighs less than 165 pounds.

We can use the same formula as before and convert 165 pounds to a z-score:

Z = (165 - 170) / 5

Next, we find the cumulative proportion corresponding to this z-score, either using a z-score table or a calculator.

Using the z-score table, we find that the cumulative proportion corresponding to Z is approximately 0.1587.

Therefore, the probability that a randomly picked student weighs less than 165 pounds is approximately 0.1587, or 15.87%.