f. The inspector will decide to shut down the restaurant if there is sufficient evidence. How strong would diners like the burden of proof to be? What about the restaurant owner? 24. Suppose you wish to test whether a coin is fair (i.e. the proportion of heads that it produces in the long run is P = 0.5). To do this, you flip a coin 15 times and count the proportion of the time that the coin is heads . b . Write the null and alternative hypotheses for testing whether this coin is fair. C. Describe intuitively how you would decide whether or not to reject the null hypothesis. d. What is a Type I error in this context ? What is a Type II error in this context ? 2. Suppose that the coin is just slightly unfair . That is, is just slightly above or just slightly below 0.50 . Comment on your ability to reject the null hypothesis given the number of flips in your sample For a = 0.05, how does an increase in the sample size affect the probability of making a Type Q5. I error ? What determines the probability of making a Type II error? Q6. Q7. A manufacturer of detergent claims that the contents of boxes sold weigh on average 16 ounces . The distribution of weight is known to be normal with a population standard deviation of 0.4 ounces. A random sample of 25 boxes is collected. a = 0.10 Using a significance level identify the range of possible sample means where you would fail to reject the null hypothesis that the population mean is 16 ounces against b . a two-sided alternative. Identify the range of sample means such that the 90% confidence intervals based on those means will contain 16 ounces Q8. A pharmaceutical manufacturer is aware that for safety issues, the impurity concentration in their pills should not exceed 3% on average . Suppose that the population standard deviation is known , and it is o = 0.4 %. A random sample of 64 pills was checked and the sample mean impurity concentration was found to be 3.07%. a . a = 0.05 Using a significance level , test the hypothesis that the population mean impurity concentration is 3% against the alternative that it is more than 3%. Identify the p-value b. for this test. Suppose that the alternative hypothesis had been two -sided rather than one -sided . Without doing any calculations, would the p-value be higher than, lower than, or the same as that found in part (a). Sketch a graph to illustrate your reasoning