Q 8.102. - Elementary Statistics

Question

Class Project: Gestation Periods of Humans. This exercise can be done individually or, better yet, as a class project. Gestation periods of humans are normally distributed with a mean of $266$ days and a standard deviation of 16 days.

a. Simulate $100$ samples of nine human gestation periods each.

b. For each sample in part (a), obtain a $95\%$ confidence interval for the population mean gestation period.

c. For the $100$ confidence intervals that you obtained in part (b), roughly how many would you expect to contain the population mean gestation period of $266$ days?

d. For the $100$ confidence intervals that you obtained in part (b), determine the number that contain the population mean gestation period of $266$ days.

e. Compare your answers from parts (c) and (d), and comment on any observed difference.

Step-by-Step Solution

Answer

Part (a) Due to chance, the answer will differ.

Part (b) $95$ percent Cl's of population mean for each of the $100$ samples in $C1-C100$ in the Session window.

Part (c) We estimate that the population mean is confined within confidence intervals of roughly $\left(100\times \frac{95}{100}\right)=95$

Part (d) Due to chance, the solution may vary.

Part (e) $\text{ The result in part (d) is exactly equal to the theoretical value obtained in part (c) }$

Part (a) Step 1: Given information

Human gestation periods have a mean of $266$ days and a standard variation of $16$ days, and are regularly distributed.

Part (a) Step 2: Explanation

We produce $100$ random samples of size $n=9$ using the MINITAB from a normal distribution with a mean of $266$ and a standard deviation of $16$

Step 1: Select$\underset{¯}{C}alc$from the menu. The Random data should be highlighted.

Step 2: Enter the data values as instructed.

Create nine rows of data.

Store in columns : $\mathrm{Cl}-\mathrm{C}100$

Mean : $266$

Standard deviation: $16$

Click: OK

Note: Answer will vary due to Randomness.

Part (b) Step 1: Explanation

Find the $95\%$ Cl's in the following way using component (a).

Step 1: Select Start from the Start menu. The basic statistics should be highlighted.

Step 2: Press 1-sample $\underset{¯}{Z}$, insert the the values provided

Tick $\square$ Sample in columns: $\mathrm{C}1-\mathrm{C}100$

$s\tan dard deviation: 16$

Step 3: Press Options, enter the following

Confidence level: $95$

Alternative: not equal

Using the procedures above, we can find 95 percent Cl's of population mean for each of the 100 samples in C1-C100 in the Session window.

Part (c) Step 1: Explanation

We estimate the population means to be contained within around $\left(100\times \frac{95}{100}\right)=95$ confidence intervals.

Part (d) Step 1: Explanation

Part (b) reveals that the population mean gestation time is 266 days in 95 confidence intervals.

Note: The answer may vary due to randomness.

Part (e) Step 1: Explanation

The theoretical value achieved in section (c) is exactly equal to the outcome in part (d).