Answer to Question #279199 in Statistics and Probability for Train

Question

Consider a manufacturing process that is known to produce bulbs that have life lengths with a standard deviation of 75 days. A potential customer will purchase bulbs from the company that manufactures the bulbs if she is convinced that the average life of the bulbs is 1550 days.

Answer 1

Consider a manufacturing process that is known to produce bulbs that have life lengths with a standard deviation of 75 days. A potential customer will purchase bulbs from the company that manufactures the bulbs if she is convinced that the average life of the bulbs is 1550 days.

1. Formulate the appropriate null and alternative hypotheses.

2. Identify situations when Type 1 and Type Il errors are committed and state their possible consequences.

3. Suppose the decision rule is "Reject Ho if a random sample of 50 bulbs has a life less than 1532 days; otherwise, fail to reject Ho." Compute for the level of significance for this test. Also, find the risk of concluding that the average is greater than 1550 days when in fact their mean score is 1500.

1.

$H_0: \mu = 1550 \\ H_1 : \mu< 1550$

The customer will not reject H₀ when $\mu = 1550$

The customer will accept H₁ if $\mu < 1550$

2.

Type I error = P(Rejecting H₀ | H₀ is true)

Type II error = P(Do not reject H₀ | H₀ is false)

3. Test-statistic

$t = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Level of significance = P(Type I error)

= P(Rejecting H₀ | H₀ is true)

$= P(\bar{x} <1532) \\ =P(Z < \frac{1532-1550}{75 / \sqrt{50}}) \\ =P(Z < -0.589) \\ = 0.2779$