Exam 12: Hypothesis Testing: Describing a Single Population

If a sample size is increased at a given $\alpha$ level, the probability of committing a Type I error increases.

The power of a test is the probability of making:

State which of the following set of hypotheses are appropriate. Explain a. Ho: µ = 25 HA: µ ≠ 25 b. Ho: µ > 25 HA: µ = 25 c. Ho: $\bar { X }$ =35 HA: $\bar { X }$ > 35 d. Ho: p = 25 HA: p ≠ 25 e. Ho: p = 0.5 HA: p > 0.5

At present, many universities in Australia are adopting the practice of having lecture recordings automatically available to students. A university lecturer is trying to investigate whether having lecture recordings available to students has significantly decreased the proportion of students passing her course. When lecture recordings were not provided to students, the proportion of students that passed her course was 80%. The lecturer takes a random sample of 25 students, when lecture recordings are offered to students, and finds that 11 students have passed the course. Is there significant evidence to support this university lecturer's claim? Use the p-value method and test at α = 0.01

For each of the following statements, state the population parameter of interest, state the appropriate null and alternative hypotheses and indicate whether the appropriate test will be a two-tail, a left tail or a right tail test. a. The average age a person registers to vote in Australia is greater than 20 years. b. A minority of office workers purchase their morning coffee. c. The average number of hours spent on a computer per day is at least 5hrs. d. The majority of students in a particular university course who attend lectures has changed from 75%, since lecture recordings have become freely available to students.

A null hypothesis is a statement about the value of a population parameter; it is put up for testing in the face of numerical evidence.

Calculate the probability of a Type II error for the following test of hypothesis: H0: μ = 50 HA: μ > 50 given that µ = 55, α = 0.05,  = 10 and n = 16.

There is a direct relationship between the power of a test and the probability of a Type II error.

We cannot commit a Type I error when the:

A social scientist claims that the average adult watches less than 26 hours of television per week. He collects data on 25 individuals' television viewing habits, and finds that the mean number of hours that the 25 people spent watching television was 22.4 hours. If the population standard deviation is known to be 8 hours, can we conclude at the 1% significance level that he is right?

In testing the hypotheses: H0: ? = 35 HA: ? < 35, The following information is known: n = 49, $\bar { X }$ = 37 and $\sigma$ = 6. The standardised test statistic equals:

Consider the hypotheses H0: μ = 950 HA: μ  950 Assume that μ = 1000,  = 200, n = 25,  = 0.10 and  = 0.6535. Recalculate $\beta$ if n is increased from 25 to 40.

In testing the hypotheses: H0:  = 500 HA:  = 500, If the value of the Z test statistic equals 2.03, then the p-value is:

The p-value of a test is the smallest value of $\alpha$ at which the null hypothesis can be rejected.

In testing the hypotheses: H0: μ = 22 HA: μ < 22 the following information was given:  = 15, n = 50, $\bar { X }$ = 17.5,  = 0.04. a. Calculate the value of the test statistic. b. Set up the rejection region. c. Determine the p-value. d. Interpret the result.

A random sample of 100 observations from a normal population whose standard deviation is 50 produced a mean of 75. Does this statistic provide sufficient evidence at the 5% level of significance to infer that the population mean is not 80?

To test the hypotheses H0: μ = 40 HA: μ  40 we draw a random sample of size 16 from a normal population whose standard deviation is 5. If we set α = 0.01, find  when µ = 37.

If a null hypothesis is rejected at the 0.05 level of significance, it cannot be rejected at the 0.10 level.

Suppose that 10 observations are drawn from a normal population whose variance is 64. The observations are: 13 21 15 19 35 24 14 18 27 30 Test at the 10% level of significance to determine whether there is enough evidence to conclude that the population mean is greater than 20.

The power of a test is the probability that it will lead us to: