Q 7.65. - Elementary Statistics

Question

Refer to Fig. $7.6$ on page 306 .

a. Why are the four graphs in Fig. 7.6(a) all centered at the same place?

b. Why does the spread of the graphs diminish with increasing sample size? How does this result affect the sampling error when you estimate a population mean, $\mu$ by a sample mean, $\tilde{x}$ ?

c. Why are the graphs in Fig. 7.6(a) bell shaped?

d. Why do the graphs in Figs. $7.6 \left(b\right)$and (c) become bell shaped as the sample size increases?

Step-by-Step Solution

Answer

Part (a) All four graphs in Figure $Fig \$7.6\$ \left(a\right)$ for various sample sizes are centered at the same location $\mu$

Part (b) when we try to estimate $\mu$ using $\overline{x}$, we can expect the value of $\overline{x}$ to be from a $\mu$nearest point, lowering the sampling error.

Part (c) The sample means $\left(\overline{x}\right)$ follow the normal distribution $\&$the curve of a normal distribution.

Part (d) The distribution of $\overline{x}$'s tends to normalcy as the sample size grows, which is why the graphs become bell-shaped.

Part (a) Step 1: Given information

The figure is 

Part (a) Step 2: Concept

Formula used: $\text{ population mean and standard deviation: }{\mu }_{\overline{x}}=\mu \text{ and }{\sigma }_{\overline{x}}=\sigma /\sqrt{n}\text{. }$

Part (a) Step 3: Explanation

Because the population variable in $Fig 7.6\left(a\right)$ is regularly distributed, and we know that sample means for normally distributed population variables are always distrusted, $\mu$ and $s.d.$, ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$

Thus, regardless of sample size, the mean of the sample means is equal to $\mu$ As a result, all four graphs in Figure $7.6$ (a) for various sample sizes are centered at the same position $\mu$

Part (b) Step 1: Explanation

We know that the sample means S.D. equals $\frac{\sigma }{\sqrt{n}}$, i.e. it is inversely proportional to the square root of the sample size $n$ As the sample size increases, the S.D. of the sample mean lowers, and the graph's spread shrinks.

We all know that standard deviation is a measure of dispersion; it tells us how far the observations are spread out or departed from the mean value. As a result, a small s.d. denotes a tiny variation from the mean value, implying that observations are strongly concentrated around the mean value.

We calculate the population mean using the sample mean $\overline{x}\&$; the mean of $\overline{x}$ is $\mu$, and the standard deviation is ${\sigma }_{\overline{x}}$ If ${\sigma }_{\overline{x}}$ drops, then At the mean $\mu$, the values of $\overline{x}$ become more concentrated. As a result, when we try to estimate $\mu$ using $\overline{x}$, we can expect the value of $\overline{x}$ to be from a $\mu$ nearest point, lowering the sampling error.

Part (c) Step 1: Explanation

The curve of a normal distribution is bell-shaped because the sample means $\left(\overline{x}\right)$ follow the normal distribution.

Part (d) Step 1: Explanation

The population variables in fig $7.6( \mathrm{b})\&7.6$ (c) do not follow a normal distribution. As a result, for small sample sizes, the sample means do not follow a normal distribution. However, for large samples, the distribution of the sample means can be approximated by the normal distribution using CLT.

As a result, the graphs in Figures 7.6(b) and 7.6(c) are not symmetric and bell-shaped for small sample sizes. The distribution of $\overline{x}$'s tends to normalcy as the sample size grows, which is why the graphs become bell-shaped.