Chapter 7.3, Problem 53E

a.

To determine

To find: The sampling distribution of $\overline{x}$ .

Explanation of Solution

Given:

  $\begin{array}{l}\text{Population mean }\left(\mu \right)=188\\ \text{Population standard deviation}\left(\sigma \right)=41\\ \text{Sample size}\left(n\right)=100\end{array}$

Calculation:

The sampling distribution of $\overline{x}$ is written as:

  $\begin{array}{l}\overline{x}\sim N\left({\mu }_{\overline{x}},{\sigma }_{\overline{x}}\right)\\ \overline{x}\sim N\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)\\ \overline{x}\sim N\left(188,\frac{41}{\sqrt{100}}\right)\\ \overline{x}\sim N\left(188,4.1\right)\end{array}$

Thus, the sampling distribution is normally distributed with mean = 100 and standard deviation = 4.1.

b.

To determine

To find: The probability that mean is within ±3 mg/dl.

Answer to Problem 53E

The probability is 0.5346.

Explanation of Solution

Calculation:

The probability that mean is within ±3 mg/dl is calculated as follows:

  $\begin{array}{c}P\left(185\le \overline{x}<191\right)=P\left(\frac{185-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{191-\mu }{\frac{\sigma }{\sqrt{n}}}\right)\\ =P\left(\frac{185-188}{\frac{41}{\sqrt{100}}}<Z<\frac{191-188}{\frac{41}{\sqrt{100}}}\right)\\ =P\left(-0.73<Z<0.73\right)\\ =0.5346\end{array}$

Thus, the required probability is 0.5346.

c.

To determine

To find: The probability that mean is within ±3 mg/dl if the sample size is 1000.

Answer to Problem 53E

The probability is 0.9792.

Explanation of Solution

Calculation:

The probability that mean is within ±3 mg/dl is calculated as follows:

  $\begin{array}{c}P\left(185\le \overline{x}<191\right)=P\left(\frac{185-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{191-\mu }{\frac{\sigma }{\sqrt{n}}}\right)\\ =P\left(\frac{185-188}{\frac{41}{\sqrt{1000}}}<Z<\frac{191-188}{\frac{41}{\sqrt{1000}}}\right)\\ =P\left(-2.31<Z<2.31\right)\\ =0.9792\end{array}$

Thus, the required probability is 0.9792.