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Reliance on solid biomass fuel for cooking and heating exposes many children from developing countries to high levels of indoor air pollution. The article "Domestic Fuels, Indoor Air Pollution, and Children's Health" (Annals of the N.Y. Academy of Sciences, 2008: 209-217) 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 755 children in biomass households, the sample mean peak expiratory flow (a person's maximum speed of expiration) was \(3.30 \mathrm{~L} / \mathrm{s}\), and the sample standard deviation was \(1.20\). For the 750 children whose households used liquefied petroleum gas, the sample mean PEF was \(4.25\) and the sample standard deviation was 1.75. a. Calculate a confidence interval at the \(95 \%\) confidence level for the population mean PEF for children in biomass households and then do likewise for children in LPG households. What is the simultaneous confidence level for the two intervals? b. Carry out a test of hypotheses at significance level \(.01\) to decide whether true average PEF is lower for children in biomass households than it is for children in LPG households (the cited article included a \(P\)-value for this test). c. \(\mathrm{FEV}_{1}\), the forced expiratory volume in 1 second, is another measure of pulmonary function. The cited article reported that for the biomass households the sample mean \(\mathrm{FEV}_{1}\) was \(2.3 \mathrm{~L} / \mathrm{s}\) and the sample standard deviation was \(.5 \mathrm{~L} / \mathrm{s}\). If this information is used to compute a \(95 \%\) CI for population mean \(\mathrm{FEV}_{1}\), would the simultaneous confidence level for this interval and the first interval calculated in (a) be the same as the simultaneous confidence level determined there? Explain.

Step by step solution

01

Calculate the Confidence Interval for Biomass Households

To calculate the confidence interval for the population mean PEF for children in biomass households, we use the formula for the confidence interval: \[CI = \bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\] Where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size, and \(z_{\alpha/2}\) is the critical value for a 95% confidence interval (approximately 1.96 for large samples). For biomass households, \(\bar{x} = 3.30\), \(s = 1.20\), and \(n = 755\). So the confidence interval is: \[3.30 \pm 1.96 \cdot \frac{1.20}{\sqrt{755}}\]Calculate the margin of error: \[1.96 \cdot \frac{1.20}{\sqrt{755}} \approx 0.086\]Thus, the 95% confidence interval for the biomass households is approximately \[(3.214, 3.386)\].

02

Calculate the Confidence Interval for LPG Households

Using the same formula as in Step 1, calculate the confidence interval for LPG households: \[CI = \bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\]For LPG households, \(\bar{x} = 4.25\), \(s = 1.75\), and \(n = 750\). The confidence interval is: \[4.25 \pm 1.96 \cdot \frac{1.75}{\sqrt{750}}\]Calculate the margin of error: \[1.96 \cdot \frac{1.75}{\sqrt{750}} \approx 0.125\]Thus, the 95% confidence interval for the LPG households is approximately \[(4.125, 4.375)\].

03

Determine Simultaneous Confidence Level for the Two Intervals

The simultaneous confidence level for two independent confidence intervals both at 95% is less than 95%. The probability that both intervals simultaneously cover their respective population parameters is approximately \(0.95^2 = 0.9025\) or roughly 90.25%.

04

Conduct Hypothesis Test for Lower Average PEF in Biomass Households

Set up the hypotheses: - Null hypothesis \(H_0: \mu_B \geq \mu_L\)- Alternative hypothesis \(H_a: \mu_B < \mu_L\)Where \(\mu_B\) is the mean PEF for biomass households and \(\mu_L\) is for LPG households.Calculate the test statistic using:\[t = \frac{(\bar{x}_B - \bar{x}_L)}{\sqrt{\frac{s_B^2}{n_B} + \frac{s_L^2}{n_L}}}\]Substitute the numbers: \[t = \frac{(3.30 - 4.25)}{\sqrt{\frac{1.20^2}{755} + \frac{1.75^2}{750}}} \approx -10.54\]With this t-value and degrees of freedom, compare with critical t-value from t-distribution table (about -2.33 for 1% significance).Since \(t < -2.33\), we reject the null hypothesis at 0.01 significance level, indicating \(\mu_B\) is significantly lower than \(\mu_L\).

05

Evaluate Simultaneous Confidence Level for Three Intervals

To evaluate if the simultaneous confidence level for three intervals is the same, calculate using: \[(0.95)^3 = 0.8574\]The simultaneous confidence level for three intervals is approximately 85.74%, which is indeed different from the simultaneous confidence level for the two confidence intervals calculated in Step 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

A hypothesis test is a method used to decide if a given assumption (hypothesis) about a population parameter should be accepted or rejected. In this case, the hypothesis test aims to determine whether the average peak expiratory flow (PEF) for children in biomass households is lower than that for children in LPG households.

To conduct the test, we establish two hypotheses:

• The **null hypothesis** (\( H_0 \)): the mean PEF of biomass household children (\( \mu_B \)) is greater than or equal to that of LPG household children (\( \mu_L \)).
• The **alternative hypothesis** (\( H_a \)): the mean PEF of biomass household children (\( \mu_B \)) is less than that of LPG household children (\( \mu_L \)).

By calculating the test statistic and comparing it against a critical value from a known distribution, here the t-distribution, we draw conclusions. If the test statistic falls in the critical region (extreme values), the null hypothesis is rejected, suggesting supporting evidence for the alternative hypothesis.

In the example given, the test statistic (\( t \approx -10.54 \)) is significantly lower than the critical t-value at a 1% significance level, resulting in the rejection of \( H_0 \) and leading to the conclusion that biomass household children have a significantly lower average PEF compared to LPG household children.

Probability

Probability serves as a measure of the likelihood that an event occurs within a given set of possible events. It ranges from 0 (impossible event) to 1 (certain event).

In hypothesis testing, probability comes into play in interpreting the confidence level and significance level:

• The **confidence level** is the probability that the interval estimate contains the true population parameter. In confidence intervals, a 95% confidence level implies there is a 95% chance that the interval includes the actual mean.
• The **significance level** (\( \alpha \)) indicates the probability of rejecting the null hypothesis when it is true, typically set as 0.05 or 0.01 in tests. This is the threshold for determining the critical value in hypothesis testing.

When calculating the simultaneous confidence level, we consider the probability that all individual confidence intervals jointly capture their respective population parameters. If two intervals are each at a 95% confidence level, then both covering their parameters simultaneously is roughly the square of 0.95, equaling 0.9025 or approximately 90.25%.

Population Mean

The population mean (\( \mu \)) is a central value describing a whole population's average. Unlike the sample mean, which is derived from a subset of the population, the population mean represents the true average across all individuals in that group.

In practical settings, especially in large populations, we often cannot measure every single individual, so we estimate the population mean using a sample.
For example, in the study of children's lung function, researchers aim to infer the mean PEF of children in biomass and LPG households using sample data from each group.

To make inferences about these population means, researchers use sample means in conjunction with statistical techniques like confidence intervals to estimate these values. This allows scientists to extrapolate findings from sample data to make broader conclusions about the population, while also accounting for possible error margins and variability.

Significance Level

The significance level (\( \alpha \)) is a crucial component in hypothesis testing, determining the criterion for rejecting a null hypothesis.

It is the probability of a Type I error—falsely rejecting the null hypothesis when it is true. Common significance levels used are 0.05 (5%) and 0.01 (1%). A lower significance level means stricter criteria for rejecting the null hypothesis, giving higher confidence in the results.

For instance, in testing if the mean PEF for biomass households is lower than for LPG households, a 1% significance level is used. This implies a very stringent test, as we only reject the null hypothesis if the evidence strongly supports the alternative.
A test statistic exceeding the critical value at this \( \alpha \) signals that results are statistically significant.

This process helps ensure conclusions drawn from the statistical test are reliable, minimizing the risk of a false positive result. Lower significance levels are thus reserved for situations where errors would have substantial consequences.