Question

Starting in the 1970s, medical technology allowed babies with very low birth weight (VLBW, less than 1500 grams, about 3.3 pounds) to survive without major handicaps. It was noticed that these children nonetheless had difficulties in school and as adults. A long-term study has followed 244 VLBW babies to age 20 years, along with a control group of 247 babies from the same population who had normal birth weight. At age 20, 188 of the VLBW group and 197 of the control group had graduated from high school. The test statistic value using the VLBW babies as group 1 is (±0.01) and the P-value for the test (±0.0001) is ________.

Answer 1

The test statistic value using the VLBW babies as group 1 is z=-2.76±0.01 and the P-value for the test (±0.0001) is 0.0030.

Explanation:

This is a hypothesis test for the difference between proportions.

The claim is that the proportion of persons with normal birth weight that graduates from high school is significantly greater than the proportion of persons with very low birth weight that graduates from high school.

Then, the null and alternative hypothesis are:

`H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2> 0`

The significance level is 0.05.

The sample 1 (VLBW group), of size n1=244 has a proportion of p1=0.77049.

`p_1=X_1/n_1=188/244=0.77049`

The sample 2 (control group), of size n2=247 has a proportion of p2=0.79757.

`p_2=X_2/n_2=197/247=0.79757`

The difference between proportions is (p1-p2)=-0.02708.

`p_d=p_1-p_2=0.77049-0.79757=-0.02708`

The pooled proportion, needed to calculate the standard error, is:

`p=(X_1+X_2)/(n_1+n_2)=(188+197)/(244+247)=(385)/(491)=0.988`

The estimated standard error of the difference between means is computed using the formula:

`s_(p1-p2)=\sqrt{(p(1-p))/(n_1)+(p(1-p))/(n_2)}=\sqrt{(0.988*0.012)/(244)+(0.988*0.012)/(247)}\\\\\\s_(p1-p2)=√(0.00005+0.00005)=√(0.0001)=0.0098`

Then, we can calculate the z-statistic as:

`z=(p_d-(\pi_1-\pi_2))/(s_(p1-p2))=(-0.02708-0)/(0.0098)=(-0.02708)/(0.0098)=-2.76`

This test is a left-tailed test, so the P-value for this test is calculated as (using a z-table):

`P-value=P(t<-2.76)=0.0030`

As the P-value (0.0030) is smaller than the significance level (0.05), the effect issignificant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of persons with normal birth weight that graduates from high school is significantly greater than the proportion of persons with very low birth weight that graduates from high school.