Q 3.95. - Elementary Statistics

Question

Days to Maturity. The first two columns of the following table provide a frequency distribution, using limit grouping, for the days to maturity of 40 short-term investments, as found in BARRON'S. The third column shows the class marks.

(a) Use the grouped-data formulas to estimate the sample mean and sample standard deviation of the days-to-maturity data. Round your final answers to one decimal place.

(b) The following table gives the raw days-to-maturity data.

Using Definitions 3.4 and 3.6 on pages 99 and 108, respectively, gives the true sample mean and sample standard deviation of the days-to-maturity data as 68.3 and 16.7, respectively, rounded to one decimal place. Compare these actual values of $\overline{x} \mathrm{and} s$ to the estimates from part (a). Explain why the grouped-data formulas generally yield only approximations to the sample mean and sample standard deviation for non-single-value grouping.

Step-by-Step Solution

Answer

Part (a) mean is 68 and the standard deviation is 16.4.

Part (b) Both outcomes are almost identical.

Part (a) Step 1. Given information.

The given table is:

Part (a) Step 2. Find the sample mean and sample standard deviation.

Days to Maturity	Frequency	Class Mark	$f_i{x}_i$	${\left(x_i-\overline{x}\right)}^2$	$f_i{\left(x_i-\overline{x}\right)}^2$
30-39	3	34.5	103.5	112.25	3366.75
40-49	1	44.5	44.5	552.25	552.25
50-59	8	54.5	436	182.25	1458
60-69	10	64.5	645	12.25	122.5
70-79	7	74.5	521.5	42.25	295.75
80-89	7	84.5	591.5	272.25	1905.75
90-99	4	94.5	378	702.25	2809
Total	$n=\sum f_i=40$		$2720$		$10510$

$$\begin{gathered} \overline{x}={\displaystyle \frac{\sum f_i{x}_i}{n}} \\ ={\displaystyle \frac{2720}{40}} \\ \overline{x}=68 \\ s=\sqrt{{\displaystyle \frac{\sum f_i\left(x_i-\overline{x}\right)}{n-1}}} \\ =\sqrt{{\displaystyle \frac{10510}{40-1}}} \\ =16.416 \\ =16.4 \end{gathered}$$

Part (b) Explanation.

The raw data of days to maturity has a mean and standard deviation of 68.3 and 16.7, respectively. These results must be compared to the results obtained in component (a). 

Although the two outcomes are nearly identical, the computing formula for s is equal to the definition both formulae provide the same answer, though rounding errors may cause changes. 

However, when completing calculations by hand, the computing formula is usually faster and easier, and it also decreases the likelihood of rounding errors.