What percentage of the general U.S. population have bachelor's degrees? The Statistical Abstract of the United States, 120th Edition, gives the percentage of bachelor's degrees by state. For convenience, the data are sorted in increasing order. $$ \begin{array}{c} 20,21,21,21,22,23,23,23,24,24,24,24,25,25,25,25,25,25,26,26,27,27,27,27,27,27 \ 27,27,28,29,29,29,29,29,29,30,30,30,30,30,31,31,32,34,34,35,35,37,38,41 \end{array} $$ (a) Which percentile is the value 30 ? (Round to the nearest whole value above.) 30 is the $$ \square $$ th percentile. (b) Find the 36 th percentile on the data. Locator $$ = $$ $$ \square $$ 36 th percentile $$ = $$ $$ \square $$ (c) Find the $$ Q_{1} $$ value on the data. Locator $$ = $$ $$ \square $$ $$ Q_{1}= $$ $$ \square $$ (d) Find the Q3 value on the data. Locator $$ = $$ $$ \square $$ $$ Q_{3}= $$ $$ \square $$ (e) Find the $$ Q_{2} $$ value on the data. $$ Q_{2}= $$ $$ \square $$ (f) Find the $$ I Q R $$ of the data. $$ I Q R= $$ $$ \square $$ (g) What are the Upper and Lower limits of the boxplot? Upper Limit $$ = $$ $$ \square $$ Lower Limit $$ = $$ $$ \square $$ (h) With five-number-summary (in the order $$ \operatorname{Min}=Q_{1}=Q_{2}=Q_{3}- $$ Max) being 20-25-27-3041 , are there any outliers? The only outlier is 20 The only outlier is 41 Outliers are 20 , and 41 . Outliers are 41, and 38 No outliers

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To solve the problem, we will follow the steps outlined for each part.

### Given Data
The data set is:
$$
\begin{array}{c}
20,21,21,21,22,23,23,23,24,24,24,24,25,25,25,25,25,25,26,26,27,27,27,27,27,27 \
27,27,28,29,29,29,29,29,29,30,30,30,30,30,31,31,32,34,34,35,35,37,38,41
\end{array}
$$

This data set contains 50 values.

### (a) Which percentile is the value 30?
To find the percentile rank of a value, we use the formula:
$$
P = \left( \frac{N_L}{N} \right) \times 100
$$
where $$ N_L $$ is the number of values less than the value of interest, and $$ N $$ is the total number of values.

1. Count the number of values less than 30:
   - The values less than 30 are: 20, 21, 21, 21, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 29, 29, 29, 29, 29, 29.
   - There are 34 values less than 30.

2. Calculate the percentile:
$$
P = \left( \frac{34}{50} \right) \times 100 = 68
$$

Thus, 30 is the **68th percentile**.

### (b) Find the 36th percentile on the data.
To find the 36th percentile, we calculate the locator:
$$
L = \frac{P}{100} \times N = \frac{36}{100} \times 50 = 18
$$
Since 18 is not a whole number, we round up to 19.

The 36th percentile is the 19th value in the ordered list:
- The 19th value is 26.

Thus, the 36th percentile is **26**.

### (c) Find the $$ Q_{1} $$ value on the data.
To find $$ Q_1 $$:
$$
L = \frac{1}{4} \times N = \frac{1}{4} \times 50 = 12.5
$$
Rounding up gives us 13.

The $$ Q_1 $$ value is the 13th value in the ordered list:
- The 13th value is 25.

Thus, $$ Q_1 = 25 $$.

### (d) Find the $$ Q_3 $$ value on the data.
To find $$ Q_3 $$:
$$
L = \frac{3}{4} \times N = \frac{3}{4} \times 50 = 37.5
$$
Rounding up gives us 38.

The $$ Q_3 $$ value is the 38th value in the ordered list:
- The 38th value is 30.

Thus, $$ Q_3 = 30 $$.

### (e) Find the $$ Q_{2} $$ value on the data.
The median $$ Q_2 $$ is found by:
$$
L = \frac{N + 1}{2} = \frac{50 + 1}{2} = 25.5
$$
Rounding up gives us 26.

The $$ Q_2 $$ value is the 26th value in the ordered list:
- The 26th value is 27.

Thus, $$ Q_2 = 27 $$.

### (f) Find the $$ IQR $$ of the data.
The interquartile range $$ IQR $$ is calculated as:
$$
IQR = Q_3 - Q_1 = 30 - 25 = 5
$$

Thus, $$ IQR = 5 $$.

### (g) What are the Upper and Lower limits of the boxplot?
The upper limit is calculated as:
$$
\text{Upper Limit} = Q_3 + 1.5 \times IQR = 30 + 1.5 \times 5 = 30 + 7.5 = 37.5
$$
The lower limit is calculated as:
$$
\text{Lower Limit} = Q_1 - 1.5 \times IQR = 25 - 1.5 \times 5 = 25 - 7.5 = 17.5
$$

Thus, the Upper Limit is **37.5** and the Lower Limit is **17.5**.

### (h) Are there any outliers?
Outliers are defined as values below the lower limit or above the upper limit.

- The minimum value is 20, which is greater than 17.5.
- The maximum value is 41, which is greater than 37.5.

Thus, the outliers are **41**.

### Summary of Answers
(a) 30 is the **68th** percentile.  
(b) 36th percentile = **26**.  
(c) $$ Q_1 = 25 $$.  
(d) $$ Q_3 = 30 $$.  
(e) $$ Q_2 = 27 $$.  
(f) $$ IQR = 5 $$.  
(g) Upper Limit = **37.5**, Lower Limit = **17.5**.  
(h) Outliers are **41**.
(a) 30 is the **68th** percentile. (b) 36th percentile = **26**. (c) $$ Q_1 = 25 $$. (d) $$ Q_3 = 30 $$. (e) $$ Q_2 = 27 $$. (f) $$ IQR = 5 $$. (g) Upper Limit = **37.5**, Lower Limit = **17.5**. (h) Outliers are **41**.

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