Question

4. The CDC publishes charts on Body Mass Index (BMI) percentiles for boys and girls of different ages. Based on the chart for girls, the mean BMI for 6-year-old girls is listed as 15.2 kg/m2. The data from which the CDC charts were developed is old and there is concern that the mean BMI for 6-year old girls has increased. The BMIs of a random sample of 30 6-year-old girls are given below.

24.5	16.3	15.7	20.6	15.3	14.5	14.7	15.7	14.4	13.2
16.3	15.9	16.3	13.5	15.5	14.3	13.7	14.3	13.7	16.0
14.2	17.3	19.5	22.8	16.4	15.4	18.2	13.9	17.6	15.5

a. State null and alternative hypotheses relevant to this situation.

b. Calculate the sample mean and standard deviation.

c. Since the sample size is relatively large, use s in place of σ and calculate the value of the z-test statistic. Then calculate the p-value.

d. Based on your answer to (c), do the sample data provide sufficient evidence that the mean BMI for 6-year-old girls has increased? Explain

Answer 1

1) Given values:

    based on given data calculated sample mean is X̅ = 16.173 and calculated standard deviation is s = 2.669 and

                the sample size is n = 30. and with known population mean is μ0 = 15.2

           (2) Our test hypothesis is :

                The following null and alternative hypothesis need to be tested,

                hypothesis =

                H0: μ = 15.2 vs

                H1: μ > 15.2

                This hypothesis corresponds to a upper-tailed, for which a t-test for one population mean.

            (3) Test statistics :

                The calculation of the t-test proceeds as follows,

                t = X̅ - μ0         16.173 - 15.2

                   --------    = ---------------   = 1.9968

                    s / √n          2.669 / 5.477

            (4) Decision about the null hypothesis :

                Since it is observed that t calculated = 1.9968, is not in the 95.0% critical value accepted range:[-∞:1.699].

                so t calculated means t = 1.9968 < t tabulated = 1.699

                it is then concluded that the null hypothesis is Rejected.

                Using the P-value approach:

                The p-value 0.027654561797876398, ( p(x≤T) = 0.0277 ). The smaller the p-value the more it supports H1.

                and since p = 0.027654561797876398 < α = 0.05

                it is then 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 population mean μ is greater than 15.2, at the 0.05 significance level.

            (6) Confidence interval:

                The 95.0% confidence interval is 15.1762 < μ < 17.1698.