Hi I need help on what is the answer for this question. Below is the picture of the problem and its references

Problem

Consequences We commit an error when we -Reject a TRUE null hypothesis (Type I error} or -Accept a FALSE null hypothesis (Type II error) Decision Matrix mm Reject Ho- Typ'e' I error Correct decision Accept Ho Correct Type II error decision 1 1-13 x 21 x... Example 1 An airport's metal detector is designed to sound of an alarm if a passenger passing through it carries a weapon. H0: The passenger does not carry a weapon Ha: The passenger carries a weapon. Example 2 A farmer was deciding whether he should apply more fertilizer to his crops or not (with re erence to the amount he has been using before) Ho: The crops do not need more fertilizer. Ha: The crops need more fertilizer Decision Rules The analysis plan includes decision rules for rejecting the null hypothesis. In practice. statisticians describe these decision rules in two ways with reference to p - value or with reference to a region of acceptance. Decision Rules P-value 'This is the probability of committing a Type I error. -Decision Rule: Reject Ho is p-valuecd. Otherwise, fail to reject Ho. Factors that Affect Power The power of a hypothesis test is affected by 3 factors... 1. Sample size (n). Other things being equal, the greater the sample size, the greater is the power of the test. 2. Signicance level (a). The higher the signicance level. the higher the power of the test. 3. The "true" value of the parameter being tested. The greater the difference between the "true" value of a parameter and the value specied in the null hypothesis. the greater the power of the test. Decision Rules Region of acceptance If the test statistic falls within the region of acceptance, the null hypothesis is not rejected. Common Critical Values a left - tailed () two- tailed (#) 0.10 Z 1.28 z 1.645 0.05 Z 1.645 z 1.96 0.01 Z 2.33 z 2.58 Level of Significance, a Claim: The population mean age is 50. a/2 + Represents critical value Ho: H = 50 a/2 H,: p # 50 Two-tail test Rejection region is shaded Ho: H S 50 H,: H > 50 Upper-tail test Ho: H 2 50 Hip 65 Example 2 Incoming freshmen are given entrance exams in different fields, including English. Over a period of years, it has been found that the average score in the English exam is 80 with a standard deviation of 8. An English instruction examines the scores for his class of 30 and finds that their average is 85. Can the instructor claim that the average score has increased at 0.01 significance level