Practice section 2 of 5

The mass, $x$ kg, of the contents of each packet in a random sample of 80 cereal packets is measured, with $\sum x=79.53$ and $\sum x^2=100.4621$.

(i) Test, at the 4% level of significance, whether the population mean mass of the contents is less than 1.10 kg.

(ii) In another test using the same data at 4%, $H_0$ states that the population mean is $\mu_0$ kg and $H_1$ states that it is not $\mu_0$ kg. Given that $H_0$ is rejected in favour of $H_1$, find the set of possible values of $\mu_0$.

In a factory, the time in minutes for an employee to install an electronics component is a normally distributed continuous random variable $T$, with standard deviation 5.0 and ordinary expected value 38.0. After background music is introduced, a sample of $n$ components is taken and the mean time for randomly chosen employees is $\bar t$ minutes. A test at 5% determines whether the population mean installation time has been reduced.

(i) State appropriate hypotheses, defining any symbols.

(ii) Given $n=50$, state the set of values of $\bar t$ for which the test rejects $H_0$.

(iii) Suppose instead that $\bar t=37.1$ and $H_0$ is not rejected. Obtain an inequality involving $n$ and find the set of values that $n$ can take.

A manufacturer makes light bulbs with power normally distributed with mean 13 watts and standard deviation 2.716 watts. A sample of 50 bulbs has mean power 13.168 watts.

(i) Test at the 5% significance level whether the manufacturer is understating the mean power.

(ii) Explain “a 5% level of significance” in this context.

(iii) At the 10% significance level, what is the least sample size for concluding that the manufacturer has understated the mean power?

For many years the Consumer Union has found that the mean lifetime of a 100-watt light bulb is 72 days of continuous use, with standard deviation 12 days. The company markets a “new and improved” bulb yielding longer life. A sample of 55 new bulbs has average lifetime 75.2 days. Does this provide sufficient evidence of superiority at the 5% significance level?

A department store reports that average spending per customer over the past six months is $215 with standard deviation $100. During a week-long sales promotion, it takes a random sample of 100 visiting customers and records their spending. It tests at 5% whether the promotion increased average spending.

(i) State, with a reason, whether the test requires assuming normally distributed customer spending.

(ii) State the null and alternative hypotheses. Given that the null hypothesis is rejected, show that the sample mean is greater than $231.45.

A random sample of 90 mobile-phone batteries has standby times $x$ hours summarized by $\sum x=3040.8$ and $\sum x^2=115773.66$. Test at 1% whether the population mean standby time is less than 36.0 hours.

The same type is advertised as having talk-time 5 hours. A separate test at 5% finds significant evidence that population mean talk-time is less than 5 hours. Using only that information, classify each statement as necessarily true, necessarily false, or neither, giving reasons:

(a) There is significant evidence at 10% that the population mean talk-time is less than 5 hours.

(b) There is significant evidence at 5% that the population mean talk-time is not 5 hours.