Practice section 3 of 5

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.

A teacher claims that the average time a student spends on mathematics revision per day is 120 minutes. A random sample of 150 students is observed and the time, $x$ min, that each student spends on mathematics revision per day noted. It is found that

$\sum(x-120)=-1500,\qquad\sum(x-120)^2=462\,000.$

(i) Test, at the 1% level of significance, whether the teacher has overstated the average time spent on mathematics revision per day by a student. [4]

(ii) Using the unbiased estimate of the population variance in (i), what is the least number of students that is required in a sample whose mean is 110 hours to arrive at a different conclusion from that in part (i)? [2]

A random sample of 50 packets of ‘Nilo Bar’ snacks is weighed and the mass, $x$ grams, is recorded. The results are summarised by

$\sum(x-150)=-250,\qquad\sum(x-150)^2=3000.$

The population mean mass is $\mu$ grams. A test is carried out at the 1% significance level with the following hypotheses:

$H_0:\mu=\mu_0,\qquad H_1:\mu\ne\mu_0.$

Given that $H_0$ is rejected in favour of $H_1$, find the set of possible values of $\mu_0$. [5]