Discussion questions

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.

A consumer association tests a battery type claimed to have mean lifetime 150 hours. A random sample of 70 lifetimes gives $\sum x=10317$, $\sum x^2=1540231$. Test $H_0:\mu=150$ against $H_1:\mu<150$: find the p-value and explain it in context.

A second random sample of 50 lifetimes gives $\sum y=7331$, $\sum y^2=1100565$. Combine the samples into one and test the same hypotheses at 10%.

Clara, who has diabetes, monitors glucose levels that vary throughout the day. A sample of 75 readings at random times over a week gives $\sum x=511.5$, $\sum x^2=4027.89$.

(i) Find the set of $\mu_0$ values for which it would be accepted that $\mu>\mu_0$ at the 10% significance level.

(ii) State, with a reason, whether that conclusion would remain valid if all 75 readings were taken during weekends.

A Mathematics test is administered to a large group. For 200 recorded scores, $\sum(x+10)=12290$ and $\sum(x+10)^2=811212$.

(i) Find unbiased estimates of the population mean and standard deviation.

(ii) Test at 5% whether the population mean score exceeds 50. State, with a reason, whether assumptions about the population distribution are needed.

The alternative hypothesis is accepted at $\alpha\%$ significance.

(iii) Find an inequality satisfied by $\alpha$.

(iv) Explain “$\alpha\%$ level of significance” in this context.

A company sells balls of string. A manager claims average length is at least 300 m. A random sample of 100 balls gives $\sum(x-300)=-60$ and $\sum(x-300)^2=1240$.

(i) Find unbiased estimates of the population mean and variance.

(ii) Test at 5% whether the manager’s claim is valid.

The process is improved and the new population variance is known to be 12.1 m². A new random sample of 100 balls has mean $k$ m. A test at 10% indicates that the manager’s claim is valid for this process.

(iii) Find the least possible $k$, giving your answer to 2 decimal places.