Chapter 17, Problem 2P

Given these data

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Determine

(a) The Mean,

(b) The standard deviation,

(c) The variance,

(d) The coefficient of variation, and

(e) The 90% confidence interval for the mean

(f) Construct a histogram. Use a range from 28 to 34 with increments of 0.4

(g) Assuming that the distribution is normal and that your estimate of the standard deviation is valid, compute the range (that is, the lower and the upper values) that encompasses 68% of the readings.

Determine whether this is a valid estimate for the data in this problem.

To determine

To calculate: The mean of the given data.

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Answer to Problem 2P

Solution:

The mean of the given data is $29.83214$.

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Calculation:

The mean of the data can be found out with the help of excel as shown below,

Step 1. First put the data in the excel as shown below,

Step 2. Use AVERAGE command in excel as shown below,

Hence, the mean of the given data is $29.83214$.

To determine

To calculate: The standard deviation of the given data.

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Answer to Problem 2P

Solution:

The standard deviation of the given data is $1.045291$.

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Calculation:

The standard deviation of the data can be found out with the help of excel as shown below,

Step 1. First put the data in the excel as shown below,

Step 2. Use STDEV command in excel as shown below,

Hence, the standard deviation of the given data is $1.045291$.

To determine

To calculate: The variance of the given data.

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Answer to Problem 2P

Solution:

The variance of the given data is $1.092632$.

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Calculation:

The variance of the data can be found out with the help of excel as shown below,

Step 1. First put the data in the excel as shown below,

Step 2. Use VAR command in excel as shown below,

Hence, the variance of the given data is $1.092632$.

To determine

To calculate: The coefficient of variation of the given data.

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Answer to Problem 2P

Solution:

The coefficient of variation of the given data is $3.50\%$.

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Formula used:

The coefficient of variation is $\frac{\text{standard}\text{deviation}}{\text{mean}}\times 100\%$.

Calculation:

The coefficient of variation of the data can be found out with the help of excel as shown below,

Step 1. First put the data in the excel as shown below,

Step 2. Use the coefficient of variation formula as shown below,

Hence, the coefficient of variation of the given data is $3.50\%$.

To determine

To calculate: The $90\%$ confidence interval for the mean of the given data.

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Answer to Problem 2P

Solution:

The $90\%$ confidence interval lower and upper value is $29.4269\text{and}\text{30}\text{.23729}$.

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Formula used:

The upper and lower range of confidence interval can be found out by,

$L=\overline{y}-\frac{s_y}{\sqrt{n}}\times t_{\frac{\alpha }{2},n-1}\text{and}U=\overline{y}+\frac{s_y}{\sqrt{n}}\times t_{\frac{\alpha }{2},n-1}$

Where $\overline{y}$ is mean, $s_y$ is standard deviation, n id number of data points, $\alpha$ is probability and $t$ is statistics calculator value.

Calculation:

Here, $\overline{y}=29.83214,s_y=1.045291,\alpha =0.10\text{and}n=28$

The t calculator value for the given values is,

$\begin{array}{c}{t}_{\frac{\alpha }{2},n-1}=t_{\frac{0.10}{2},28-1}\\ =t_{0.05,27}\\ =2.051\end{array}$

Use the formula to calculate the lower range,

$\begin{array}{c}L=\overline{y}-\frac{s_y}{\sqrt{n}}\times t_{\frac{\alpha }{2},n-1}\\ =29.83214-\frac{1.045291}{\sqrt{28}}\times 2.051\\ =29.83214-0.40515\\ =29.4269\end{array}$

And the upper range is,

$\begin{array}{c}U=\overline{y}+\frac{s_y}{\sqrt{n}}\times t_{\frac{\alpha }{2},n-1}\\ =29.83214+\frac{1.045291}{\sqrt{28}}\times 2.051\\ =29.83214+0.40515\\ =30.23729\end{array}$

Hence, the $90\%$ confidence interval lower and upper value is $29.4269\text{and}\text{30}\text{.23729}$.

To determine

To graph: The histogram of the given data by using a range from $28\text{to}\text{34}$ with intervals of $0.4$

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Graph:

The histogram of the data can be found out with the help of excel as shown below,

Step 1. First put the data in the excel as shown below,

Step 2. Now insert the bin array range as shown below,

Step 3. Select on DATA, Data Analysis, Histogram and then OK.

Step 4. Enter the input, Bin and output range as shown below,

Step 5. The following result will appear.

Step 6. Now click on the insert and then column chart.

Interpretation:

It can be concluded from the histogram that 30 has repeated maximum time in the given data.

To determine

To calculate: The $68\%$ lower and upper range value of the given data if the calculated standard deviation is valid and then decide whether this is a valid estimate for the data.

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Answer to Problem 2P

Solution:

The $68\%$ lower and upper range value is $28.78685\text{and}\text{30}\text{.87743}$ and this is not a valid estimate for the data.

Explanation of Solution

Given Information:

The data is,

29.65	28.55	28.65	30.15	29.35	29.75	29.25
30.65	28.15	29.85	29.05	30.25	30.85	28.75
29.65	30.45	29.15	30.45	33.65	29.35	29.75
31.25	29.45	30.15	29.65	30.55	29.65	29.25

Calculation:

As the $68\%$ falls between $\overline{y}-s_y$ and $\overline{y}+s_y$ that is,

$29.83214-1.04529=28.78685$

And,

$29.83214+1.04529=30.87743$

From the Histogram, the data within this range $28.78685\text{to}\text{30}\text{.87743}$ can be calculated as 22.

Thus, the percentage of data in this range is,

$\frac{22}{28}=78.6\%$

Which is higher than $68\%$. So, this is not a valid estimate.