Question:

a.) In 1965, a newspaper carried a story about a high school student who reported getting 9207 heads and 8743 tails in 17,950 coin tosses. Is this a significant discrepancy from the null hypothesis \[{{H}_{0}}:p=1/2\]

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b.) Jack Youden, a statistician at the National Bureau of Standards, contacted the student and asked him exactly how the performed the experiment. To save time the student tossed groups of five coins at a time, and a younger brother recorded the result, shown in the following table:

Are the data consistent at the 1% significance with the hypothesis that all the coins are fair (p=1/2)?

Solution: (a)The following null and alternative hypotheses are:

\[\begin{aligned} & {{H}_{0}}:p=\text{0}\text{.5} \\ & {{H}_{A}}:p\ne \text{0}\text{.5} \\ \end{aligned}\]

This test corresponds to a two tailed z-test for proportions. The significance level is set at

\[\alpha =\text{0}\text{.05}\]

The critical value for the given significance level is computed as

\[{{z}_{c}}=\text{1}\text{.96}\]

The rejection region of the null hypothesis is \(R=\left\{ z:\,\,|z|>{{z}_{c}} \right\}=\text{1}\text{.96}\). Now, the sample proportion is

\[\hat{p}=\frac{\text{9207}}{\text{17950}}=\text{0}\text{.513}\]

Using all this information, the z-statistics is computed as

\[z=\frac{\hat{p}-p}{\sqrt{\frac{p\left( 1-p \right)}{n}}}=\frac{\text{0}\text{.513}-\text{0}\text{.5}}{\sqrt{\frac{\text{0}\text{.5}\left( 1-\text{0}\text{.5} \right)}{\text{17950}}}}=\text{3}\text{.463}\]

The two-tailed critical z-value for \(\alpha =\text{0}\text{.05}\) is given by \({{z}_{c}}=\text{1}\text{.96}\). Since we have that \(|z|=\text{3}\text{.463}\) > \({{z}_{c}}=\text{1}\text{.96}\), then we reject the null hypothesis.

This means that we have enough evidence to support the claim that the coin is not fair, at the 0.05 significance level.

(b) The following table shows the expected probabilities under the assumption that the coin is fair