A sample of thirty automobiles was tested for fuel efficiency, in miles per gallon (mpg). The following frequency distribution was obtained. Round your answer to 1 decimal place.

\begin{tabular}{|l|l|}
\hline
Class Boundary & Frequency \\
\hline
[tex]$7.5-12.5$[/tex] & 3 \\
\hline
[tex]$12.5-17.5$[/tex] & 5 \\
\hline
[tex]$17.5-22.5$[/tex] & 15 \\
\hline
[tex]$22.5-27.5$[/tex] & 5 \\
\hline
[tex]$27.5-32.5$[/tex] & 2 \\
\hline
\end{tabular}

The estimated median value of the distribution is:

Answer: [tex]$\square$[/tex]



Answer :

To find the estimated median value of the given frequency distribution of fuel efficiency in miles per gallon, follow these steps:

1. List the class boundaries and their frequencies:
- 7.5 - 12.5: 3
- 12.5 - 17.5: 5
- 17.5 - 22.5: 15
- 22.5 - 27.5: 5
- 27.5 - 32.5: 2

2. Calculate the cumulative frequencies:
- Cumulative frequency up to 7.5 - 12.5: 3
- Cumulative frequency up to 12.5 - 17.5: 3 + 5 = 8
- Cumulative frequency up to 17.5 - 22.5: 8 + 15 = 23
- Cumulative frequency up to 22.5 - 27.5: 23 + 5 = 28
- Cumulative frequency up to 27.5 - 32.5: 28 + 2 = 30

3. Find the total number of observations, [tex]\( n \)[/tex]:
- [tex]\( n = 3 + 5 + 15 + 5 + 2 = 30 \)[/tex]

4. Determine the position of the median in the ordered data set:
- Median position [tex]\( = \frac{n + 1}{2} = \frac{30 + 1}{2} = 15.5 \)[/tex]

5. Identify the median class:
- The median position, 15.5, falls within the cumulative frequency of 23, which corresponds to the class boundary 17.5 - 22.5.

6. Extract the necessary values for the median class:
- Lower boundary ([tex]\(L\)[/tex]) of the median class = 17.5
- Frequency ([tex]\(f\)[/tex]) of the median class = 15
- Cumulative frequency before the median class ([tex]\(cf_{prev}\)[/tex]) = 8
- Class interval size ([tex]\(h\)[/tex]) = 22.5 - 17.5 = 5

7. Calculate the median using the formula:

[tex]\[ \text{Median} = L + \left( \frac{\frac{n+1}{2} - cf_{prev}}{f} \right) \times h \][/tex]

Substituting the values:

[tex]\[ \text{Median} = 17.5 + \left( \frac{15.5 - 8}{15} \right) \times 5 \][/tex]

8. Simplify the calculations:

[tex]\[ \text{Median} = 17.5 + \left( \frac{7.5}{15} \right) \times 5 \][/tex]
[tex]\[ \text{Median} = 17.5 + \left( 0.5 \right) \times 5 \][/tex]
[tex]\[ \text{Median} = 17.5 + 2.5 \][/tex]
[tex]\[ \text{Median} = 20.0 \][/tex]

Thus, the estimated median value of the distribution is [tex]\( \boxed{20.0} \)[/tex].