A study is conducted regarding shatterproof glass used in automobiles. Twenty-six glass panes are coated with an anti-shattering film. Then a 5-pound metal ball is fired at 70mph at each pane. Five of the panes shatter. We wish to determine whether, in the population of all such panes, the probability the glass shatters under these conditions is different from π = 0.2.

(a) State the appropriate null and alternative hypotheses.
(b) Check the conditions for trusting the conclusion of the test, and calculate the observed value of an appropriate test statistic.
(c) Calculate the rejection region and draw a conclusion, given the significance level α = 0.05.
(d) Calculate the p-value.
(e) Compute the power of the test if the true π was in fact 0.3.

(a) The appropriate null and alternative hypotheses are:

Null hypothesis (H0): The probability the glass shatters under these conditions is equal to π = 0.2.
Alternative hypothesis (Ha): The probability the glass shatters under these conditions is different from π = 0.2.

(b) To check the conditions for trusting the conclusion of the test, we need to verify that the requirements for conducting a two-proportion z-test are met:

1. Randomness: The glass panes should be selected randomly to ensure that the sample is representative of the entire population.
2. Independence: Each glass pane should be independent from the others. This means that the shattering of one pane does not influence the shattering of another.
3. Success-Failure Assumption: Both the success (shattering) and failure (not shattering) categories should have enough observations. The general rule is to have at least 10 successes and 10 failures.

In this case, we can assume that the conditions are met since the panes were coated randomly, the shattering of one pane does not affect the others, and we have 5 shattered panes and 21 intact panes, meeting the requirement of having at least 10 successes and 10 failures.

The observed value of an appropriate test statistic can be calculated using the two-proportion z-test formula:

z = (p1 - p2) / sqrt((p_hat * (1 - p_hat) / n1) + (p_hat * (1 - p_hat) / n2))

where:
p1 = proportion of shattered panes in the sample
p2 = hypothesized proportion (π = 0.2)
p_hat = (x1 + x2) / (n1 + n2) (proportion of shattered panes in the pooled sample)
n1 = sample size of shattered panes
n2 = sample size of intact panes

In this case, p1 = 5/26, p2 = 0.2, n1 = 5, n2 = 21. Calculate p_hat and substitute the values into the formula to obtain the observed test statistic.

(c) To calculate the rejection region and draw a conclusion, we need to determine the critical values based on the significance level α = 0.05. The rejection region will be two-tailed because the alternative hypothesis is two-sided. We can use a standard normal distribution table or a statistical software to find the critical z-values.

(d) The p-value can be calculated by finding the probability of observing a test statistic as extreme as the observed value, assuming the null hypothesis is true. For a two-sided test, the p-value is the probability of observing a test statistic as extreme as the observed value in both tails of the distribution. This can be obtained using a standard normal distribution table or a statistical software.

(e) To compute the power of the test, we need to know the true proportion (π = 0.3) under the alternative hypothesis. Using this value, we can calculate the power of the test using the formula:

Power = 1 - β

where β is the probability of making a Type II error (failing to reject the null hypothesis when it is false). The value of β can be calculated by determining the area under the null distribution curve that falls within the rejection region. Again, this can be done using a standard normal distribution table or statistical software.

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