Chapter 6.4, Problem 1DC

Solution

To explain: increasing the sample size in each trial affects the shape of the histogram.

  

Comparing it with data sets of different sample size shows that the spread of the data decreases as the sample size increases. However, the shape and center of the two histograms appear to be relatively similar.

Given information:

It is given that in each trial count the number of decimals is less than 0.6 out of 80 random decimals.

Concept Used:

Biased questions:

Questions that are flawed in a way that leads to inaccurate results are called biased questions. Questions may be biased in several ways:

The wording of the question may encourage or pressure the respondent to answer in a particular way.

The question may be perceived as too sensitive to answer truthfully.

The question may not provide the respondent with enough information to give an accurate opinion.

Calculation:

Use the random number generator to obtain 80 random decimals between 0 and 1.

  

Now, determine the number of decimals with value less than 0.6.

From the above sample table of values, the number of decimals less than 0.6 is 55.

Now, determine the sample proportion.

  $\begin{array}{c}\text{Sample}\text{Proportion}=\frac{55}{80}\\ =0.6875\end{array}$

Use the random number generator to obtain 40 random decimals between 0 and 1.

  

Create a histogram of the sample proportions with intervals of 0.05.

  

The shape of the histogram with a sample size of 80 is approximately a normal distribution with a center at the interval 0.55-0.65.

The data from the histogram for the sample proportions with sample size of 80 ranges from 0.45 to 0.70.

  

The shape of the histogram with a sample size of 40 is approximately a normal distribution with a center at the interval 0.60-0.65.

The data from the histogram for the sample proportions with a sample size of 40 ranges from 0.60 to 0.65.

From the observations on the range of the two histograms, it follows that the spread of the data decreases as the sample size increases.

However, the shape remains to be approximately a normal distribution even if the same size is increased.

Also, the center of the histogram of the two cases appears to be relatively close around the interval 0.55-0.65.