Question

Table 2.76 gives the percent of children under five considered to be underweight.

What is the best estimate for the mean percentage of underweight children? What is the standard deviation? Which interval(s) could be considered unusual? Explain.

Short answer

• Estimate of mean percentage of underweight students is sample mean $=26.573$
• Sample standard deviation
• Interval $43.25-48.7$is unusual.

Step-by-step solution

To find

Best estimate of mean percentage of underweight students and standard deviation and also determine unusual intervals.

Explanation

The percentage class of underweight children and their corresponding class frequencies.

The best estimate of mean percentage of underweight children is sample mean. The formula to find sample mean from frequency table is

Sample mean $\overline{x}=\frac{\sum f{x}_m}{n}$

Where $\mathrm{f}=$class frequency,

$x_m=$class midpoint and

$\mathrm{n}=$total frequency

The formula to find standard deviation for grouped frequency data is

$\mathrm{s}=\sqrt{\frac{\sum {f_{x_m}}^2}{n}-{\overline{x}}^2}$

To find sample mean and standard deviation

Consider the table is :

Calculated the Sample value

$\begin{array}{r}\overline{x}=\frac{\sum f{x}_m}{n}\\ =\frac{1328.65}{50}\\ =26.573\end{array}$

Sample mean = 26.573

To calculate standard deviation the formula is shown below:

$s=\sqrt{\frac{\sum f{x}^2}{n}-{\overline{x}}^2}$

Substitute values as shown below:

$$\begin{gathered} s=\sqrt{\frac{\sum f{x}^2}{n}-{\overline{x}}^2} \\ \begin{array}{r}=\sqrt{\frac{38880.02}{50}-(26.573{)}^2}\\ =\sqrt{777.6004-706.1243}\\ =8.4544\end{array} \end{gathered}$$

If the probability of the relevant interval occuring is less than $0.05, the interval is said to be unusual. Each interval's probability is calculated by dividing the class frequency by n=50.

The probability for interval $(43.25-48.7)$is 0.02 which is$<0.05.$

Hence interval $43.25-48.7$is said to be unusual.