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.

Short answer

a. No

b.$p-value=0$

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

Step-by-step solution

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.

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 $max-Q3andQ1-min$. So, there is no outliers.

Explanation (part b)

If the mean daily calcium intake for women $18to24$years of age is really $1200mg$, then the likelihood of getting a sample of 36 women with a mean intake of $856.2$mg or smaller is roughly .

$p-value=0$

Explanation (part c)

State: $H_0:\mu =1200versus{H}_a:\mu <1200$, where $\mu$is the actual mean daily calcium intake of women $18to24$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 $1200mg$of calcium daily, on average.