You might need to references the Z-table to solve the following questions.

## Sample questions

For the following questions, consider that in a population of adults ages 18 to 65, BMI (body mass index) is normally distributed with a mean of 27 and a standard deviation of 5.

What is the BMI score for which half of the population has a lower value?

**Answer:**27In this case, using intuition is very helpful. If you have a normal distribution for the population, then half of the values lie below the mean (because it's symmetrical and the total percentage is 100%). Here, the mean is 27, so 50%, or half, of the population of adults has a BMI lower than 27.

What BMI marks the bottom 25% of the distribution for this population?

**Answer:**23.65You want to find the value of

*X*(BMI) where 25% of the population lies below it. In other words, you want to find the 25th percentile of*X.*First, you need to find the 25th percentile for*Z*(using the*Z-*table) and then change the*z*-value to an*x-*value by using the*z-*formula:To find the 25th percentile for

*Z*(or the cutoff point where 25% of the population lies below it), look at the*Z-*table and find the probability that's closest to 0.25.The probabilities for the

*Z-*table are the values*inside*the table. The numbers on the outsides that tell which row/column you're in are actual*z-*values, not probabilities.Searching the Z-table, you see that the closest probability to 0.25 is 0.2514.

Next, find what

*z-*score this probability corresponds to. After you've located 0.2514 inside the table, find its corresponding row (–0.6) and column (0.07). Put these numbers together and you get the*z-*score of –0.67. This is the 25th percentile for*Z.*In other words, 25% of the*z-*values lie below –0.67.To find the corresponding BMI that marks the 25th percentile, use the

*z-*formula and solve for*x.*You know thatSo 25% of the population has a BMI lower than 23.65.

What BMI marks the bottom 5% of the distribution for this population?

**Answer:**18.80You want to find the value of

*X*(BMI) where 5% of the population lies below it. In other words, you want to find the 5th percentile of*X.*First, you need to find the 5th percentile for*Z*(using the*Z-*table) and then change the*z*-value to an*x-*value by using the*z-*formula:To find the 5th percentile for

*Z*(or the cutoff point where 5% of the population lies below it), look at the*Z-*table and find the probability that's closest to 0.05.You see that the closest probability to 0.05 is either 0.0495 or 0.0505 (use 0.0505 in this case).

Next, find what

*z-*score this probability corresponds to. After you've located 0.0505 inside the table, find its corresponding row (–1.6) and column (0.04). Put these numbers together and you get the*z-*score of –1.64. This is the 5th percentile for*Z.*In other words, 5% of the*z-*values lie below –1.64.To find the corresponding BMI that marks the 5th percentile, use the

*z-*formula and solve for*x.*So 5% of the population has a BMI lower than 18.80.

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