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 go through each part step by step.

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

### Total Number of Data Points
There are 50 data points in total.

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

**Count 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 35 values less than 30.

**Calculating the percentile:**
$$
P = \left( \frac{35}{50} \right) \times 100 = 70
$$
Thus, 30 is the $$ 70 $$th percentile.

### (b) Find the 36th percentile on the data.
To find the 36th percentile, we calculate the locator:
$$
L = \frac{P}{100} \times (N + 1) = \frac{36}{100} \times (50 + 1) = 36 \times 0.51 = 18.36
$$
We round up to the nearest whole number, which gives us 19.

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

### (c) Find the $$ Q_1 $$ value on the data.
$$ Q_1 $$ is the 25th percentile, so we calculate:
$$
L = \frac{25}{100} \times (50 + 1) = 25 \times 0.51 = 12.75
$$
Rounding up gives us 13.

**Finding $$ Q_1 $$:**
The 13th value is 25.

### (d) Find the $$ Q_3 $$ value on the data.
$$ Q_3 $$ is the 75th percentile:
$$
L = \frac{75}{100} \times (50 + 1) = 75 \times 0.51 = 38.25
$$
Rounding up gives us 39.

**Finding $$ Q_3 $$:**
The 39th value is 35.

### (e) Find the $$ Q_2 $$ value on the data.
$$ Q_2 $$ is the median (50th percentile):
$$
L = \frac{50}{100} \times (50 + 1) = 50 \times 0.51 = 25.5
$$
Rounding gives us 26.

**Finding $$ Q_2 $$:**
The average of the 25th and 26th values:
$$
\frac{27 + 27}{2} = 27
$$

### (f) Find the $$ IQR $$ of the data.
$$
IQR = Q_3 - Q_1 = 35 - 25 = 10
$$

### (g) What are the Upper and Lower limits of the boxplot?
- **Lower Limit:** $$ Q_1 - 1.5 \times IQR = 25 - 1.5 \times 10 = 10 $$
- **Upper Limit:** $$ Q_3 + 1.5 \times IQR = 35 + 1.5 \times 10 = 50 $$

### (h) Are there any outliers?
- Outliers are defined as values below the lower limit or above the upper limit.
- The minimum value is 20 and the maximum is 41.
- Since 20 is above 10 and 41 is below 50, there are no outliers.

### Summary of Results
- (a) 30 is the $$ 70 $$th percentile.
- (b) 36th percentile $$ = 26 $$ (Locator $$ = 19 $$).
- (c) $$ Q_1 = 25 $$ (Locator $$ = 13 $$).
- (d) $$ Q_3 = 35 $$ (Locator $$ = 39 $$).
- (e) $$ Q_2 = 27 $$.
- (f) $$ IQR = 10 $$.
- (g) Upper Limit $$ = 50 $$, Lower Limit $$ = 10 $$.
- (h) No outliers.
- (a) 30 is the 70th percentile. - (b) 36th percentile = 26 (Locator = 19). - (c) $$ Q_1 = 25 $$ (Locator = 13). - (d) $$ Q_3 = 35 $$ (Locator = 39). - (e) $$ Q_2 = 27 $$. - (f) $$ IQR = 10 $$. - (g) Upper Limit = 50, Lower Limit = 10. - (h) No outliers.

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