Question

The file HeadCirc.csv contains head circumference measurements made on female and male babies at age two months. The measurements were made in an infant health centre serving a large suburban district when the babies were presented for routine 2-month health checks in a particular week. All the babies were from single births (i.e., no twins or triplets, etc). The variables are – Circ Head circumference in centimetres (cm) Sex The sex of the baby (“Female” or “Male”) (a) Provide a graphical display that allows the distribution of head circumference among male babies to be compared with the distribution among female babies. [5] (b) Report on the evidence that average head circumference differs between females and males at two months of age in this district. Follow all the steps required in a hypothesis test, including especially the statement and evaluation of assumptions. If you think there are problems with the assumptions, proceed as if any problems have been (or could be) resolved. [12] (c) Obtain a 99% confidence interval for the difference in mean head circumference between female and male babies aged 2 months, and state the meaning of this interval in its realworld context.

Answer 1

(a)

The distributions can be compared by plotting histograms.

First we construct a frequency table as below,

	Male	Female
34-36	2	1
36-38	0	3
38-40	5	14
40-42	29	27
42-44	14	5

Now, using histograms we observe the distribution

(b)

The sample means are shown below:

Also, the sample standard deviations are:

and

the sample sizes

and

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations can be used.

(2) Rejection Region

Based on the information provided, the significance level is = 0.99, and the degrees of freedom are df = 98.

Assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is , for and df = 98.

The rejection region for this two-tailed test is .

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

= 3.343

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0012, and since p = 0.0012 <= 0.99, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected.

Therefore, there is enough evidence to claim that average head circumference differs between females and males at two months of age in this district, at the 0.99 significance level.

(c)

Confidence Interval

Confidence interval for means is given by:

= 1.05

Pooled variance

= 2.4659

Substituting we get,

Lower bound =

= 1.046

Upper bound =

= 1.046

The 99% confidence interval is