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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

4.38

4.80

5.02

4.57

4.74

5.19

4.87

4.76

4.56

4.68

​(a) Determine a point estimate for the population mean.

A point estimate for the population mean is

​(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.There is a 95% chance that the true mean pH of rain water is between ___ and ___

B.There is 95​% confidence that the population mean pH of rain water is between ___ and ___

C.If repeated samples are​ taken,95​% of them will have a sample pH of rain water between ___ and ___

​(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​% chance that the true mean pH of rain water is between___ and___

B.If repeated samples are​ taken, 99​% of them will have a sample pH of rain water between___

and ___.

C.There is 99​% confidence that the population mean pH of rain water is 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 ___(decreases / increases).

This makes sense since the ___ (margin of error / sample size/ point estimate) ____(increases as well / decreases as well).

Respuesta :

Answer:

a) Point estimate of the population mean = 4.883

b) B.There is 95​% confidence that the population mean pH of rain water is between 4.646 and 5.120.

c) C.There is 99​% confidence that the population mean pH of rain water is between 4.549 and 5.217.

d) As the level of confidence​ increases, the width of the interval increases.

This makes sense since the margin of error increases as well.

Step-by-step explanation:

We have a sample for the pH of rain.

The mean of the sample is:

[tex]M=\dfrac{1}{12}\sum_{i=1}^{12}(5.3+5.72+4.38+4.8+5.02+...+4.56+4.68)\\\\\\ M=\dfrac{58.59}{12}=4.883[/tex]

The sample standard deviation is:

[tex]s=\sqrt{\dfrac{1}{(n-1)}\sum_{i=1}^{12}(x_i-M)^2}\\\\\\s=\sqrt{\dfrac{1}{11}\cdot [(5.3-4.883)^2+(5.72-4.883)^2+...+(4.68-4.883)^2]}\\\\\\[/tex]

[tex]s=\sqrt{\dfrac{1}{11}\cdot [(0.17)+...+(0.1)+(0.04)]}\\\\\\s=\sqrt{\dfrac{1.526625}{11}}=\sqrt{0.1387841}\\\\\\s=0.373[/tex]

a) The point estimation for the population mean is the sample mean and has a value of 4.883.

b) We have to calculate a 95% confidence interval for the mean.

The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.

The sample mean is M=4.883.

The sample size is N=12.

When σ is not known, s divided by the square root of N is used as an estimate of σM:

[tex]s_M=\dfrac{s}{\sqrt{N}}=\dfrac{0.373}{\sqrt{12}}=\dfrac{0.373}{3.464}=0.108[/tex]

The t-value for a 95% confidence interval is t=2.201.

The margin of error (MOE) can be calculated as:

[tex]MOE=t\cdot s_M=2.201 \cdot 0.108=0.237[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=M-t \cdot s_M = 4.883-0.237=4.646\\\\UL=M+t \cdot s_M = 4.883+0.237=5.12[/tex]

The 95% confidence interval for the mean is (4.646, 5.120).

B.There is 95​% confidence that the population mean pH of rain water is between 4.646 and 5.120.

c) We have to calculate a 99% confidence interval for the mean.

The t-value for a 99% confidence interval is t=3.106.

The margin of error (MOE) can be calculated as:

[tex]MOE=t\cdot s_M=3.106 \cdot 0.108=0.334[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=M-t \cdot s_M = 4.883-0.334=4.549\\\\UL=M+t \cdot s_M = 4.883+0.334=5.217[/tex]

The 99% confidence interval for the mean is (4.549, 5.217).

C.There is 99​% confidence that the population mean pH of rain water is between 4.549 and 5.217.

d) When the confidence level is increased, the width also increases as it has to include more possible values for the true mean of the population.

As the level of confidence​ increases, the width of the interval increases.

This makes sense since the margin of error increases as well.

Based on the information given, it should be noted that the point estimate of the population mean is 4.883.

Also, there is 95% confidence that the population mean pH of rain water is between 4.646 and 5.120.

Furthermore, there is 99% confidence that the population mean pH of rain water is between 4.549 and 5.217.

Lastly, the increase in the level of confidence will lead to an increase in the width of the interval.

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