Questions: The following data represent the pH of rain for a random sample of 12 rain dates. A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot indicates there are no outliers. Complete parts a) through d) below. 5.20 5.72 4.99 4.80 5.02 4.58 4.74 5.19 4.61 4.76 4.56 5.68 A. If repeated samples are taken, 95% of them will have a sample pH of rain water between and 1. B. There is a 95% probability that the true mean pH of rain water is between and 1. C. There is 95% confidence that the population mean pH of rain water is between 4.73 and 5.24. (c) Construct and interpret a 99% confidence interval for the mean pH of rainwater. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to two decimal places as needed.) A. There is a 99% probability that the true mean pH of rain water is between and . B. There is 99% confidence that the population mean pH of rain water is between 4.62 and 5.36. C. If repeated samples are taken, 99% of them will have a sample pH of rain water between and . (d) What happens to the interval as the level of confidence is changed? Explain why this is a logical result. As the level of confidence increases, the width of the interval This makes sense since the

The following data represent the pH of rain for a random sample of 12 rain dates. A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot indicates there are no outliers. Complete parts a) through d) below. 5.20 5.72 4.99 4.80 5.02 4.58 4.74 5.19 4.61 4.76 4.56 5.68 A. If repeated samples are taken, 95% of them will have a sample pH of rain water between and 1. B. There is a 95% probability that the true mean pH of rain water is between and 1. C. There is 95% confidence that the population mean pH of rain water is between 4.73 and 5.24. (c) Construct and interpret a 99% confidence interval for the mean pH of rainwater. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to two decimal places as needed.) A. There is a 99% probability that the true mean pH of rain water is between and . B. There is 99% confidence that the population mean pH of rain water is between 4.62 and 5.36. C. If repeated samples are taken, 99% of them will have a sample pH of rain water between and . (d) What happens to the interval as the level of confidence is changed? Explain why this is a logical result. As the level of confidence increases, the width of the interval This makes sense since the
Transcript text: The following data represent the pH of rain for a random sample of 12 rain dates. A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot indicates there are no outliers. Complete parts a) through d) below. 5.20 5.72 4.99 4.80 5.02 4.58 4.74 5.19 4.61 4.76 4.56 5.68 A. If repeated samples are taken, $95 \%$ of them will have a sample pH of rain water between $\square$ and $\square$ 1. B. There is a $95 \%$ probability that the true mean pH of rain water is between $\square$ and $\square$ 1. C. There is $95 \%$ confidence that the population mean pH of rain water is between 4.73 and 5.24 . (c) Construct and interpret a $99 \%$ confidence interval for the mean pH of rainwater. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to two decimal places as needed.) A. There is a $99 \%$ probability that the true mean pH of rain water is between $\square$ and $\square$ . B. There is $99 \%$ confidence that the population mean pH of rain water is between 4.62 and 5.36 . C. If repeated samples are taken, $99 \%$ of them will have a sample pH of rain water between $\square$ and $\square$ . (d) What happens to the interval as the level of confidence is changed? Explain why this is a logical result. As the level of confidence increases, the width of the interval $\square$ This makes sense since the
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Solution

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Solution Steps

Step 1: Calculate Sample Statistics

The sample mean (\(\bar{x}\)) and sample standard deviation (\(s\)) for the pH of rainwater are calculated as follows:

\[ \bar{x} = 4.9875 \] \[ s = 0.3980 \] The sample size (\(n\)) is: \[ n = 12 \]

Step 2: Construct 95% Confidence Interval

To construct the 95% confidence interval for the true mean, we use the formula:

\[ \bar{x} \pm t \frac{s}{\sqrt{n}} \]

Where \(t\) is the critical t-value for \(n-1 = 11\) degrees of freedom at a 95% confidence level, which is approximately \(2.2\). Thus, the confidence interval is calculated as:

\[ 4.9875 \pm 2.2 \cdot \frac{0.3980}{\sqrt{12}} = (4.73, 5.24) \]

Step 3: Construct 99% Confidence Interval

For the 99% confidence interval, we again use the same formula, but with the critical t-value for \(n-1 = 11\) degrees of freedom at a 99% confidence level, which is approximately \(3.11\):

\[ 4.9875 \pm 3.11 \cdot \frac{0.3980}{\sqrt{12}} = (4.63, 5.34) \]

Final Answer

The results for the confidence intervals are as follows:

  • The 95% confidence interval for the true mean pH of rainwater is \((4.73, 5.24)\).
  • The 99% confidence interval for the true mean pH of rainwater is \((4.63, 5.34)\).

Thus, the final answers are:

  • For part B: \(\boxed{(4.73, 5.24)}\)
  • For part C: \(\boxed{(4.63, 5.34)}\)
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