Hypothesis tests：練習題

In a specific region, the number of emails received by an office follows a Poisson distribution with a mean of 12 per day. Following a change in office policy, it is observed that on one randomly chosen day, the office received only 6 emails. Test at the 5% significance level whether the mean number of emails received per day has decreased. Which of the following is the correct conclusion of this test?

The number of flaws per meter in a length of fabric follows a Poisson distribution with mean \(\lambda\). A manufacturer claims that \(\lambda = 0.8\). To test this claim at the \(5\%\) significance level, a random sample of 5 meters is inspected and 8 flaws are found in total. Formulate the null and alternative hypotheses to test if the mean number of flaws has increased.

A manufacturer claims that the mean weight of a component is 150g. The weights are normally distributed with a standard deviation of 8g. A hypothesis test is conducted with a random sample of 25 components to test \(H_0: \mu = 150\) against \(H_1: \mu < 150\). The null hypothesis is rejected if the sample mean weight, \(\bar{x}\), is less than 147.36g.
Calculate the probability of a Type I error, and the probability of a Type II error if the true population mean is 148g.

Find the critical value \( z \) for a one-tailed hypothesis test (lower tail) of a population mean at a 1% significance level, assuming a normal distribution.

For a two-tailed hypothesis test regarding a population mean, a researcher calculates a test statistic of \( z = -1.90 \). Determine the p-value for this test.

A researcher performs a one-tailed test for a mean with known standard deviation \( \sigma = 12 \) at the 1% significance level. Find the minimum sample size needed so that the power of the test is 0.95 when the true mean is 4 units higher than the null hypothesis mean.

A lightbulb manufacturer claims that the mean lifespan of their bulbs is 800 hours. It is known that the lifespan is normally distributed with a standard deviation of 40 hours. A consumer protection agency tests a random sample of 50 bulbs and finds the sample mean lifespan to be 788 hours. Test, at the 5% significance level, whether there is evidence to suggest that the mean lifespan is less than 800 hours.

A manufacturer claims that the average weight of a particular brand of chocolate bar is 50.0 grams. A consumer rights group suspects that the machine is under-filling the bars and decides to carry out a hypothesis test. They take a random sample of 60 bars and find that the sample mean weight is 49.6 grams. It is known from previous data that the standard deviation of the weights of these bars is 1.5 grams.

(a) State the null and alternative hypotheses for this test.
(b) Carry out the test at the 2.5% significance level and state your conclusion clearly.
(c) Calculate the p-value for this test and explain how it supports your conclusion in part (b).