Reliance on solid biomass fuel for cooking and heating exposes many children fro
ID: 3313633 • Letter: R
Question
Reliance on solid biomass fuel for cooking and heating exposes many children from developing countries to high levels of indoor air pollution. An article presented information on various pulmonary characteristics in samples of children whose households in India used either biomass fuel or liquefied petroleum gas (LPG). For the 751 children in biomass households, the sample mean peak expiratory flow (a person's maximum speed of expiration) was 3.7 L/s, and the sample standard deviation was 1.2. For the 759 children whose households used liquefied petroleum gas, the sample mean PEF was 4.28 and the sample standard deviation was 1.98. a) Calculate a confidence interval at the 95% confidence level or the population mean PEF for children in biomass households and then do likewise for children in LPG households Round your answers to two decimal places. biomass households LPG households What is the simultaneous confidence level for the two intervals? (Round your answer to two decimal places.) (b) Carry out a test of hypotheses at significance vel0.01 to decide whether true average PEF is lower for children in biomass households than it is for children in LPG households (the artice included a P-value for this test). (Round your test statistic to two decimal places and your P-value to four decimal places.) State the relevant hypotheses. O Ho: biomass-LPG-0 Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.) P-value = State the conclusion in the problem context. Reject Ho. The data does not suggest that the true average PEF is lower for children in biomass households than it is for children in LPG households. O Fail to reject Ho. The data does not suggest that the true average PEF is lower for children in biomass households than it is for children in LPG households O Fail to reject Ho. The data suggests that the true average PEF is lower for children in biomass households than it is for children in LPG households. O Reject Ho. The data suggests that the true average PEF is lower for children in biomass households than it is for children in LPG households. (e) FEV1, the forced expiratory volume in 1 second, is another measure of pulmonary function. The cited artice reported that for the biomass households the sample mean FEV1 was 2.7 Lys and the sample standard deviation was 0.5L/s If this information is used to compute a 95% CI or population mean FEV1 would the simultaneous confidence evel or this interval and the first interval calculated in a be the same as the simultaneous confidence level determined there? Explain Select-.The two confidence intervals in (a) were from twoSelect- random samples, and so the confidence levels could be multiplied: Set same group of 751 children Selec constitute independent random samples. Hence, wesay that the two resulting Cls for that population have the same simultaneous confidence level found in part (a). However, two variables (PEF and FEV1) collected on theExplanation / Answer
The 95% confidence interval for mean : x ± z*standard error
Where z is the multiplier for 95% = 1.96
Standard error = standard deviation/n
The 95% confidence interval for mean of biomass households : 3.7 ± 1.96*1.2/751
The 95% confidence interval for mean of biomass households: (3.696868, 3.703132)
The 95% confidence interval for mean of LPG households : 4.28 ± 1.96*1.98/759
The 95% confidence interval for mean of LPG households: (4.274887, 4.285113)
The simultaneous confidence level for two intervals is (0.95*0.95) = 0.9025 =90.25%
Hypotheses:
H0 : µbiomass - µLPG = 0
H1 : µbiomass - µLPG < 0
z = ( x1 - x2)/[s21/n1 + s22/n2] = (3.7-4.28)/ [1.22/751 + 1.982/759]
=-6.89175
p-value is found using r software:
> pnorm(-6.89175) #P-value
[1] 2.755508e-12
Conclusion:
Since, p-value < 0.05,
Reject H0. The data suggests that the true average PEF if lower for children in biomass households than it is for children in LPG households.
No, the two confidence intervals in (a) were two independent random samples, and so the confidence levels could be multiplied: 90.25%. However, two variables (PEF and FEV1 ) collected on the same group of 751 children does not constitute independent random samples. Hence, we cannot say that the two resulting CIs for the population have the same simultaneous confidence level found in (a).