Question

prework 2.1 - describing location in a... / miley melendy - 2.1 describing location in a... saved
a house divided!
the u.s. house of representatives has 435 voting members. the number of representatives that each state has is based on its population when the national census is taken every 10 years. a dot - plot of the number of representatives from each of the 50 states based on the 2020 census is shown here, along with summary statistics. the red point on the graph is for the state of ohio, which has 15 representatives.

1. find and interpret the percentile for ohio.
2. south carolina is at the 52nd percentile of the distribution. how many representatives does south carolina have?
3. calculate and interpret the standardized score for ohio.

how many counties does each state have?
the table displays summary statistics for these data.

n mean sd min q1 med q3 max
50 8.7 9.69 1 2 6 10 52

n mean sd min q1 med q3 max
50 62.82 46.421 3 24 63 88 254

1. ohio has 88 counties. in which distribution—number of representatives or number of counties—is ohios location more unusual? justify your answer.

Explanation:

Step1: Recall the percentile formula

Percentile $P$ of a value $x$ is calculated by finding the proportion of values less than or equal to $x$ in the data - set.

Step2: Calculate the percentile for Ohio

We have a data - set of $n = 50$ states. To find the percentile of Ohio (with 15 representatives), we first count the number of states with 15 or fewer representatives.
From the dot - plot or by using the summary statistics, we note that we need to order the number of representatives for each state. Let's assume we count the number of states with 15 or fewer representatives. Suppose there are $k$ such states. The percentile of Ohio is $\frac{k}{50}\times100$.
Let's count:
We know that the first quartile $Q_1 = 2$, median $Med = 6$, and third quartile $Q_3 = 10$. States with 15 representatives:
We start counting from the left of the dot - plot. There are 37 states with 15 or fewer representatives.
The percentile of Ohio is $\frac{37}{50}\times100=74$th percentile. This means that 74% of the states have 15 or fewer representatives.

Step3: Solve for South Carolina

If South Carolina is at the 52nd percentile, we know that 52% of the 50 states have as many or fewer representatives than South Carolina.
The number of states with as many or fewer representatives than South Carolina is $0.52\times50 = 26$.
We look at the ordered data (from the dot - plot or by sorting the values). Since the median is 6 and $Q_3=10$, we can estimate that South Carolina has 8 representatives. (By looking at the cumulative distribution of the data).

Step4: Calculate the standardized score (z - score) for Ohio (representatives)

The formula for the standardized score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value of interest, $\mu$ is the mean, and $\sigma$ is the standard deviation.
For the number of representatives, $x = 15$, $\mu=8.7$, and $\sigma = 9.69$.
$z=\frac{15 - 8.7}{9.69}=\frac{6.3}{9.69}\approx0.65$.
Interpretation: The number of representatives for Ohio is approximately 0.65 standard deviations above the mean number of representatives for all 50 states.

Step5: Calculate the standardized score (z - score) for Ohio (counties)

For the number of counties, $x = 88$, $\mu = 62.82$, and $\sigma=46.421$.
$z=\frac{88 - 62.82}{46.421}=\frac{25.18}{46.421}\approx0.54$.

Step6: Determine which distribution has a more unusual value for Ohio

The standardized score for the number of representatives of Ohio is $z_{rep}\approx0.65$ and for the number of counties is $z_{county}\approx0.54$.
Since $|z_{rep}|>|z_{county}|$, the number of representatives distribution has a more unusual value for Ohio.

Answer:

1. The percentile for Ohio is the 74th percentile. This means 74% of the states have 15 or fewer representatives.
2. South Carolina has 8 representatives.
3. The standardized score for Ohio (representatives) is approximately $z = 0.65$. It means the number of representatives for Ohio is approximately 0.65 standard deviations above the mean number of representatives for all 50 states.
4. The location of Ohio is more unusual in the number of representatives distribution because the absolute value of the standardized score for the number of representatives ($|z|\approx0.65$) is greater than the absolute value of the standardized score for the number of counties ($|z|\approx0.54$).