Q. 73 - The Practice of Statistics for AP

Question

Healthy bones The recommended daily allowance (RDA) of calcium for women between the ages of $18$ and $24$ years is $1200$ milligrams (mg). Researchers who were involved in a large-scale study of women’s bone health suspected that their participants had significantly lower calcium intakes than the RDA. To test this suspicion, the researchers measured the daily calcium intake of a random sample of $36$ women from the study who fell in the desired age range. The Minitab output below displays descriptive statistics for these data, along with the results of a significance test.

(a) Determine whether there are any outliers. Show your work.

(b) Interpret the P-value in context.

(c) Do these data give convincing evidence to support the researchers’ suspicion? Carry out a test pg $571$ to help you answer this question.

Step-by-Step Solution

Answer

a. No

b. $p-value = 0$

c.  It appears that women in this age range are getting less than $1200 mg$of calcium daily, on average.

Step 1: Given information

The recommended daily allowance (RDA) of calcium for women between the ages of $18$ and $24$ years is $1200$ milligrams (mg).

To test this suspicion, the researchers measured the daily calcium intake of a random sample of $36$ women from the study who fell in the desired age range.

Step 2: Explanation (part a)

The first quartile $Q1$, which represents a quarter of the way through the list of all data. $Q_1=632.3$

The third quartile $Q3$, which represents three-quarters of the way through the list of all data. There is no third quartile. $Q_3=1090.5$

Therefore, $IQR=458.2$ which is greater than $max - Q3 and Q1 - min$. So, there is no outliers.

Step 3: Explanation (part b)

If the mean daily calcium intake for women $18 to 24$ years of age is really $1200 mg$, then the likelihood of getting a sample of 36 women with a mean intake of $856.2$ mg or smaller is roughly .

$p-value = 0$

Step 4: Explanation (part c)

State: $H_0 : \mu = 1200 versus H_a : \mu < 1200$, where $\mu$ is the actual mean daily calcium intake of women $18 to 24$ years of age.

Plan: One-sample t-test for $\mu$.

Random: The sample was randomly selected. 

Normal: The sample size was $36$, which is at least $30$.

Independent: There are clearly many more than $360$ women in the United States. 

Do: $t =-6.73$, P-value is approximately $0$.

Conclude: Since our P-value is less than $0.05,$ we reject $H_0$.

It appears that women in this age range are getting less than $1200 mg$ of calcium daily, on average.