Question

1. old folks here is a stemplot of the percents of residents aged 65 and older in the 50 states:

7|0
8|8
9|8
10|019
11|16777
12|01122456778999
13|0001223344455689
14|023568
15|24
16|9
key. 15|2 means 15.2% of this states residents are 65 or older
(a) find the percentile for colorado, where 10.1% of the residents are aged 65 and older
(b) rhode island is at the 80th percentile of the distribution. interpret this value. what percent of rhode islands residents are aged 65 and older?

Explanation:

Step1: Count total data points

There are 50 data - points (since we are considering 50 states).

Step2: Find position for Colorado

Arrange the data in ascending order. Count the number of data - points less than 10.1. There are 4 data - points less than 10.1 (7.0, 8.8, 9.8, 10.0). The percentile $P$ is calculated as $P=\frac{\text{Number of data points less than the value}}{\text{Total number of data points}}\times100$. So for Colorado, $P=\frac{4}{50}\times100 = 8$.

Step3: Interpret 80th percentile for Rhode Island

The 80th percentile means that 80% of the states have a lower percentage of residents aged 65 and older than Rhode Island. To find the value corresponding to the 80th percentile, we use the formula $i=\frac{p}{100}\times n$, where $p = 80$ and $n = 50$. So $i=\frac{80}{100}\times50=40$. Since $i$ is an integer, the 80th percentile is the average of the 40th and 41st ordered data - values. Counting the data points in the stemplot, the 40th value is 13.8 and the 41st value is 13.9. The average is $\frac{13.8 + 13.9}{2}=13.85$.

Answer:

(a) 8th percentile
(b) Interpretation: 80% of the states have a lower percentage of residents aged 65 and older than Rhode Island. The percentage of Rhode Island's residents aged 65 and older is 13.85%