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The likelihood of exactly \(x\) successes on \(n\) repeated trials in an experiment with two alternative outcomes is known as binomial probability (commonly called a binomial experiment).

The binomial probability is\({}_n{C_x} \cdot {p^x} \cdot {(1 - p)^{n - x}}\)if the likelihood of success on an individual trial is\(p\).

The number of alternative combinations of \(x\) objects chosen from a set of \(n\) objects is indicated by \({}_n{C_x}\).

(a)

Let the given be:

\(\begin{array}{c}p = 0.5\\n = 20\\\alpha = 0.05\\{H_0}:p = 0.5\\{H_a}:p > 0.5\end{array}\)

The number of success X within the sample of \(20\) with constant probability of success follows the binomial distribution: \(X \sim b(20,0.5)\)

If the null hypothesis is true, the P-value is the chance of getting the test statistic's value, or a value more extreme. The probability to the right of \(x\) is hence the P-value.

If the P-value is less than the \(0.05\) significance level, the null hypothesis will be rejected.

Because the last column of the table does not include the precise number \(0.05\), the significance level of \(0.05\) cannot be achieved.

The maximum \(\alpha \) less than \(0.05\) that can be achieved is the value in the table's last column that is less than \(0.05\): \(\alpha = 0.0207\)

(b)

Let the given be:

\(\begin{array}{c}p = 0.5\\n = 20\\\alpha = 0.0207\\{H_0}:p = 0.5\\{H_a}:p > 0.5\end{array}\)

The number of success X within the sample of \(20\) with constant probability of success follows the binomial distribution: \(X \sim b(20,0.5)\)

If the null hypothesis is true, the P-value is the chance of getting the test statistic's value, or a value more extreme. The probability to the right of \(x\) is hence the P-value.

If the P-value is less than or equal to the significance level of \(0.0207\), the null hypothesis is rejected. If \(x > 14\), reject the null hypothesis.

Now, \({p_A} = 60\% = 0.60\).

When the null hypothesis is false, the probability of making a type II error is the probability of not rejecting the null hypothesis. Using the \(p = 0.6\) definition of binomial probability, we obtain:

\(\begin{aligned}{c}\beta (0.60) &= P(X \le 14)\\ &= P(X = 0) + P(X = 1) + ... + P(X = 14)\\ &= 0.8744\\ &= 87.44\% \end{aligned}\)

Now, \({p_A} = 80\% = 0.80\).

When the null hypothesis is false, the probability of making a type II error is the probability of not rejecting the null hypothesis. Using the \(p = 0.6\) definition of binomial probability, we obtain:

\(\begin{aligned}{c}\beta (0.80) &= P(X \le 14)\\ &= P(X = 0) + P(X = 1) + ... + P(X = 14)\\ &= 0.1958\\ &= 19.58\% \end{aligned}\)

If the null hypothesis is true, the P-value is the chance of getting the test statistic's value, or a value more extreme. The probability to the right of x is hence the P-value.

If the P-value is less than or equal to the significance level of \(0.0207\), the null hypothesis is rejected. If \(x > 14\), reject the null hypothesis. Hence, fail to reject the null hypothesis \({H_0}\).