Question

question 10
59 randomly selected students were asked how many siblings were in their family. let x represent the number of pairs of siblings in the students family. the results are as follows:

of siblings | 0 | 1 | 2 | 3 | 4 | 5 | 6

frequency | 9 | 8 | 5 | 7 | 9 | 13 | 8

round all your answers to 4 decimal places where possible.

the mean is:
the median is:
the sample standard deviation is:
the first quartile is:
the third quartile is:
what percent of the respondents have at least 3 siblings? %
38% of all respondents have fewer than how many siblings?

hint: using a frequency table
video on using the ti84 for statistics given a frequency table +

Explanation:

Mean Calculation

Step1: Calculate total of (siblings * frequency)

We multiply each number of siblings by its frequency and sum them up.
For \( x = 0 \), \( 0\times9 = 0 \)
For \( x = 1 \), \( 1\times8 = 8 \)
For \( x = 2 \), \( 2\times5 = 10 \)
For \( x = 3 \), \( 3\times7 = 21 \)
For \( x = 4 \), \( 4\times9 = 36 \)
For \( x = 5 \), \( 5\times13 = 65 \)
For \( x = 6 \), \( 6\times8 = 48 \)
Now sum these products: \( 0 + 8 + 10 + 21 + 36 + 65 + 48 = 188 \)

Step2: Calculate total frequency

Sum the frequencies: \( 9 + 8 + 5 + 7 + 9 + 13 + 8 = 59 \)

Step3: Calculate mean

Mean \( \bar{x}=\frac{\sum (x\times f)}{\sum f}=\frac{188}{59}\approx3.1864 \)

Step1: Find the position of the median

The total number of observations \( n = 59 \). The median is the \( \frac{n + 1}{2}=\frac{59+ 1}{2}=30^{th} \) value.

Step2: Cumulative frequency

We calculate cumulative frequencies:

• For \( x = 0 \), cumulative frequency \( CF = 9 \)
• For \( x = 1 \), \( CF=9 + 8=17 \)
• For \( x = 2 \), \( CF = 17+5 = 22 \)
• For \( x = 3 \), \( CF=22 + 7=29 \)
• For \( x = 4 \), \( CF=29+9 = 38 \)

The \( 30^{th} \) value falls in the class where \( x = 4 \) (since \( 29<30\leq38 \))

Step1: Calculate \( \sum (x^{2}\times f) \)

• For \( x = 0 \), \( 0^{2}\times9 = 0 \)
• For \( x = 1 \), \( 1^{2}\times8 = 8 \)
• For \( x = 2 \), \( 2^{2}\times5 = 20 \)
• For \( x = 3 \), \( 3^{2}\times7 = 63 \)
• For \( x = 4 \), \( 4^{2}\times9 = 144 \)
• For \( x = 5 \), \( 5^{2}\times13 = 325 \)
• For \( x = 6 \), \( 6^{2}\times8 = 288 \)

Sum these: \( 0+8 + 20+63 + 144+325 + 288 = 848 \)

Step2: Calculate sample variance \( s^{2}=\frac{\sum (x^{2}\times f)-\frac{(\sum (x\times f))^{2}}{n}}{n - 1} \)

We know \( \sum (x\times f)=188 \), \( n = 59 \)
\( \frac{(\sum (x\times f))^{2}}{n}=\frac{188^{2}}{59}=\frac{35344}{59}\approx599.0508 \)
\( \sum (x^{2}\times f)-\frac{(\sum (x\times f))^{2}}{n}=848 - 599.0508 = 248.9492 \)
\( s^{2}=\frac{248.9492}{58}\approx4.2922 \)

Step3: Calculate sample standard deviation \( s=\sqrt{s^{2}} \)

\( s=\sqrt{4.2922}\approx2.0717 \)

Answer:

\( 3.1864 \)

Median Calculation