Question

A South African mathematician, John Kerrich, was visiting Copenhagen in 1940 when Germany invaded Denmark. Kerrich was forced to spend the next five years in an internment camp, and to pass the time, he carried out a series of experiments. One such experiment involved flipping a coin 10,000 times and keeping track how many heads he obtained. Of all the 10,000 coin flips, 5067 came up heads.

a.Use the normal approximation to calculate a 95% confidence interval for the true probability of heads for Kerrich’s coin, and interpret your result.

b.Use an exact method to calculate a 95% confidence interval for the true probability of heads for Kerrich’s coin (you will need a computer), and interpret your result.

c.Compare your results from a and b. Why do the results look so similar? What would have to happen in order for these results to look substantially different?

d.Do you think the coin he used in this experiment was fair? Explain.

Answer 1

Here we want to obtain 95% confidence interval for population proportion ( P )

Let' write given information.

n = sample size = 10000

Number of success = x = 5067

Confidence level = c = 95%

a) Let's used minitab to construct confidence interval for population proportion ( P )

Step 1) Click on Stat>>>Basic statistics>>>1-proportion...

Step 2) select summarized data

number of events = x = 10000

Number of trials = n = 5067

Step 3) click on option

The given confidence level is = 95.0

so put "Confidence level " = 95.0

Alternative = not equal

then click on "Use test and interval based on normal approximation"

Then click on OK and again click on OK

So we get the following output

Since 0.5 lies in the above confidence interval the coin is a fair coin.

b)

Let's used minitab to construct confidence interval for population proportion ( P )

Step 1) Click on Stat>>>Basic statistics>>>1-proportion...

Step 2) select summarized data

number of events = x = 10000

Number of trials = n = 5067

Step 3) click on option

The given confidence level is = 95.0

so put "Confidence level " = 95.0

Alternative = not equal

Then click on OK and again click on OK

So we get the following output

The exact 95% confidence interval is  (0.496850, 0.516546)

Since 0.5 lies in the above confidence interval the coin is a fair coin.

c) Both the confidence intervals include 0.5 and so the interpretation is same based on both the confidence interval.

Note that the width of confidence interval of part a ) is  0.516499-0.496901=0.0196

And  the width of confidence interval of part b ) is =0.49685- 0.516546= 0.0197

This is because the sample size n = 10000 is very large and also = 5067/10000 = 0.5067 is close to 0.5.

d) Yes because 0.5 included in the confidence interval.