Question

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. 5.30 5.72 4.99 4.80 5.02 4.57 4.74 5.19 5.29 4.76 4.56 4.91 ​(a) Determine a point estimate for the population mean. A point estimate for the population mean is nothing. ​(Round to two decimal places as​ needed.) ​(b) Construct and interpret a 95​% 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. If repeated samples are​ taken, 95​% of them will have a sample pH of rain water between nothing and nothing. B. There is a 95​% probability that the true mean pH of rain water is between nothing and nothing. C. There is 95​% confidence that the population mean pH of rain water is between nothing and nothing. ​(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 nothing and nothing. B. There is 99​% confidence that the population mean pH of rain water is between nothing and nothing. C. If repeated samples are​ taken, 99​% of them will have a sample pH of rain water between nothing and nothing. ​(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 ▼ increases. decreases. This makes sense since ▼ including more numbers for consideration makes it more likely one of them is correct. all confidence intervals of a given level of confidence have the same width. including fewer numbers for consideration makes it more likely one of them is correct. Click to select your answer(s).

Answer 1

The data is provided as follows:

Data
5.3
5.72
4.99
4.8
5.02
4.57
4.74
5.19
5.29
4.76
4.56
4.91

(a)

The sample size is n=12. The provided sample data along with the data required to compute the sample mean shown in the table below:

	X	X2
	5.3	28.09
	5.72	32.7184
	4.99	24.9001
	4.8	23.04
	5.02	25.2004
	4.57	20.8849
	4.74	22.4676
	5.19	26.9361
	5.29	27.9841
	4.76	22.6576
	4.56	20.7936
	4.91	24.1081
Sum =	59.85	299.781

The sample mean is computed as follows:

This is also the point estimate of the population mean.

Also

(b)

We need to construct the 95% confidence interval for the population mean μ. The following information is provided:

Sample Mean =	4.9875
Sample Standard Deviation (s) =	0.340991202
Sample Size (n) =	12

The critical value for α=0.05 and df=n−1=11 degrees of freedom is . The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the population mean is 4.77<μ<5.20

C. There is 95​% confidence that the population mean pH of rain water is between 4.77 and 5.20

(c)

The critical value for α=0.01 and df=n−1=11 degrees of freedom is . The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 99% confidence interval for the population mean is 4.68<μ<5.29

B. There is 99​% confidence that the population mean pH of rain water is between 4.68 and 5.29

(d)

As the level of confidence​ increases, the width of the interval increases. This makes sense since including more numbers for consideration makes it more likely one of them is correct.

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