Question

6. For H{0} / μ = 90 Hi(): µ #90 a) Determine the P-value if overline x = 90.48 . n = 7 and σ = 3 . b) Determine the probability of type II error if µtrue = 92. (α = 0.05) Ans.: 0.674486, 0.57926

Answer 1

a) The P-value is 0.674486 when the sample mean
`(\(\overline{x}\))` is 90.48, the sample size
`(\(n\))` is 7, and the population standard deviation
`(\(\sigma\))`is 3.

b) The probability of a type II error is 0.57926 when the true population mean
`(\(\mu_(true)\))`is 92, given a significance level
`(\(\alpha\))` of 0.05.

Step-by-step explanation:

a) To find the P-value for a hypothesis test when
`\(H_0: \mu` =
`90\)` and
`\(H_1: \mu \\eq 90\)`, use the formula for the t-test statistic:
`\(t` =

`{x} - \mu}{(\sigma)/(√(n))}\)`. Given
`\(\overline{x}` =
`90.48\)`,
`\(n = 7\)`, and
`\(\sigma = 3\)`, calculate
`\(t\)`. Then, using a t-distribution table or software, find the probability that
`\(|t|\)` exceeds the calculated value to determine the P-value.

b) For a type II error calculation, consider the true population mean
`(\(\mu_(true)` =
`92\))` under
`\(H_1\)`when
`\(H_0\)` is rejected. With a given significance level
`(\(\alpha = 0.05\))`, determine the critical value(s) for rejection and then find the probability of not rejecting
`\(H_0\)` when
`\(\mu\)`is actually
`\(\mu_(true)\)`. This probability represents the probability of a type II error.

Understanding these probabilities (P-value and type II error) is crucial in hypothesis testing. The P-value indicates the strength of evidence against the null hypothesis, and a type II error occurs when the null hypothesis is not rejected when it should be. Both are essential concepts in statistical inference, allowing researchers to make informed decisions based on collected data.