4. The CDC publishes charts on

4. The CDC publishes charts on BodyMass Index (BMI) percentiles for boys and girls of different ages.Based on the chart for girls, the mean BMI for 6-year-old girls islisted as 15.2 kg/m2. The data from which the CDC charts weredeveloped is old and there is concern that the mean BMI for 6-yearold girls has increased. The BMIs of a random sample of 306-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 thissituation.

b. Calculate the sample mean and standard deviation.

c. Since the sample size is relatively large, uses in place of σ and calculatethe value of the z-test statistic. Then calculate thep-value.

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

Answer:

1) Given values:

    based on given data calculated sample mean isX̅ = 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 betested,

               hypothesis =

               H0: μ = 15.2 vs

               H1: μ > 15.2

               This hypothesis corresponds to a upper-tailed, for which a t-testfor 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 the95.0% 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 smallerthe 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 0.05significance level.

           (6) Confidence interval:

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


 
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