• A Level Maths

Demo

• A Level
• Hypothesis Tests (Normal)

1. The average packet of crisps weighed 50g with a standard deviation of 5g in 2010. A student believes that crisp packets are smaller now and wishes to test this at the 10% significance level.

1. Write down the null and alternate hypotheses for the test.
2. The student buys 10 packets of crisps and finds that the mean mass is 48g. Complete the test.

1. $\mathrm{H}_0: \mu = 50$
   $\mathrm{H}_1: \mu < 50$ where $\mu$ is the mean weight of crisps
2. $X \sim N(50, \frac{5^2}{10})$
   $\mathrm{P}(X < 48) = 0.103 > 0.1$
   Reject $\mathrm{H}_1$. There is insufficient evidence to suggest the weight of crisps has reduced.

2. A school claims that, on average, 60% of their pupils pass their GCSEs each year, with a standard deviation of 8%. A pupil believes that the number is lower, and wishes to test this at the 5% significance level.

1. Write down the null and alternate hypotheses for the test.
2. The student collects data over 5 years and finds that the mean % of students passing their GCSEs is 55%. Complete the test.

1. $\mathrm{H}_0: \mu = 60$
   $\mathrm{H}_1: \mu < 60$ where $\mu$ is the average percentage of pupils passing GCSEs
2. $X \sim N(60, \frac{8^2}{5})$
   $\mathrm{P}(X < 55) = 0.0811 > 0.05$
   Reject $\mathrm{H}_1$. There is insufficient evidence to suggest the percentage of pupils passing is lower.

3. A new diet claims that its followers lose an average of 10 kg in 3 months, with a standard deviation of 2 kg. In a study of 10 followers of the diet, they lost an average of 8.9 kg in 3 months. Test, at the 5% level of significance, whether the claimed weight loss on the diet is too high.

$\mathrm{H}_0: \mu = 10$
$\mathrm{H}_1: \mu < 10$ where $\mu$ is the average weight lost by followers of the diet in 3 months
$X \sim N(10, \frac{2^2}{10})$
$\mathrm{P}(X < 8.9) = 0.0410 < 0.05$
Reject $\mathrm{H}_0$. There is evidence to suggest the mean weight lost is less than 10 kg.

4. The weights of bags of sweets are normally distributed with mean 120g and standard deviation 11g. Ten bags are sampled and their mean weight is found to be 126g. Determine, at the 5% level of significance, whether there is evidence that the mean weight of the bags is over 120 g.

$\mathrm{H}_0: \mu = 120$
$\mathrm{H}_1: \mu > 120$ where $\mu$ is the mean weight of bags of sweets
$X \sim N(120, \frac{11^2}{10})$
$\mathrm{P}(X > 126) = 0.0423 < 0.05$
Reject $\mathrm{H}_0$. There is evidence to suggest the mean weight of bags of sweets is more than 120g.

5. The fuel consumption of average cars on the road is normally distributed with a mean of 40 mpg and standard deviation of 3 mpg. The fuel consumption of 25 brand new cars have a mean of 41 mpg. Determine, at the 5% level of significance, whether there is evidence that new cars are more fuel efficient.

$\mathrm{H}_0: \mu = 40$
$\mathrm{H}_1: \mu > 40$ where $\mu$ is the mean mpg of cars
$X \sim N(40, \frac{3^2}{25})$
$\mathrm{P}(X > 41) = 0.0478 < 0.05$
Reject $\mathrm{H}_0$. There is evidence to suggest new cars are more fuel efficient.

6. The volume of water in bottles of water have a mean of 500 ml and standard deviation of 20 ml. The volume of 40 bottles was measured and the mean was 496 ml. Determine, at the 10% level of significance, whether these bottles are underfilled.

$\mathrm{H}_0: \mu = 500$
$\mathrm{H}_1: \mu < 500$ where $\mu$ is the mean volume of water in bottles
$X \sim N(500, \frac{20^2}{40})$
$\mathrm{P}(X < 496) = 0.103 > 0.1$
Reject $\mathrm{H}_1$. There is insufficient evidence to suggest the bottles are underfilled.

7. A student wants to test whether the average price of school lunches has increased across the country at the 10% significance level. 10 years ago, the price was normally distributed with mean £4, with a variance of 50p.

1. State 2 assumptions the student must make in order to conduct the test.
2. The student conducts the test and obtains a p-value of $0.852$. What can the student conclude?

1. The current price is still normally distributed, and the variance of the current prices is still 50p.
2. The p-value is greater than 10%. There is insufficient evidence to suggest prices have increased.

8. The average height of oak trees is 3.7 m with a standard deviation of 0.3 m. It was found that the average height of 12 oak trees growing in a garden is 3.9 m. The gardener wishes to test whether their trees were taller than average at the 2% significance level.

1. Find the critical region(s) for this test.
2. Write down the conclusion of the test.

1. One tailed test, $X \sim N(3.7, \frac{0.3^2}{12})$
   $\mathrm{P}(X > n) = 0.02 \Rightarrow n = 3.88$
   Critical region: $X > 3.88$
2. In the critical region, so reject $\mathrm{H}_0$
   There is evidence to suggest the gardener's trees are taller.

9. The average amount of time children spend playing video games is 3.2 hours on weekdays, with a standard deviation of 1.2 hours. A teacher wants to test if children spend more time playing video games on Saturdays, and asks 5 children how long they spend playing video games on Saturdays, getting the following results (in hours): $$3.7, 4.0, 4.1, 4.3, 4.9$$ Test, at the 2.5% significance level, whether children spend more time playing video games on Saturdays than on weekdays.

$\mathrm{H}_0: \mu = 3.2$
$\mathrm{H}_1: \mu > 3.2$ where $\mu$ is the amount of time children spend playing video games
$X \sim N(3.2, \frac{1.2^2}{5})$
$\mathrm{P}(X > 4.2) = 0.0312 > 0.025$
Reject $\mathrm{H}_1$. There is insufficient evidence to suggest children spend more time playing video games on Saturdays.

10. According to the manufacturer, their phone batteries last a mean of 6.2 hours and have a variance of 2.7 hours². The lifetimes 50 phones were measured and the mean was 6.1 hours. A phone user wishes to test this claim at the 10% level of significance.

1. Find the critical region(s) for this test.
2. Write down the conclusion of the test.

1. Two tailed test, $X \sim N(6.2, \frac{2.7}{50})$
   $\mathrm{P}(X > n) = 0.05 \Rightarrow n = 6.29$
   Critical regions: $X > 6.29$ and $X < 6.11$
2. $6.1$ is in the critical region, so reject $\mathrm{H}_0$.
   There is evidence to suggest the manufacturer's claim is wrong.

11. A manufacturer claims that the amount of cereal in their boxes is normally distributed with mean 1000g and variance 120g. A student bought 6 boxes of cereal and found that they weighed (in g): $$952, 967, 970, 972, 977, 982$$ Determine, at the 10% level of significance, whether the amount of cereal in these boxes match the amount claimed by the manufacturer.

$\mathrm{H}_0: \mu = 1000$
$\mathrm{H}_1: \mu \neq 1000$ where $\mu$ is the amount of cereal in boxes
$X \sim N(1000, \frac{120}{6})$
$\mathrm{P}(X < 970) = 0.0668 > 0.05$
Reject $\mathrm{H}_1$. There is insufficient evidence to suggest amount of cereal in these boxes differs from the manufacturer's claim.

12. It is claimed that the the average number of people in a class is normally distributed with mean 30 and variance 5. A student believes that the average class is bigger than this, and wishes to test this claim at the 2.5% significance level.

1. Write down the null and alternate hypotheses.
2. A sample of 10 classes had an average of 31 people per class. Complete the test.
3. The student collects a different sample and found that the mean is still 31 people per class. Given that they can reject the null hypothesis, find the minimum sample size they could have used.

1. $\mathrm{H}_0: \mu = 30$
   $\mathrm{H}_1: \mu > 30$ where $\mu$ is the number of people in a class
2. $X \sim N(30, \frac{5}{10})$
   $\mathrm{P}(X > 31) = 0.0228 < 0.025$
   Reject $\mathrm{H}_0$. There is evidence to suggest class sizes are bigger than claimed.
3. With 9 classes in the sample, the probability becomes $0.0359$ which is no longer in the critical region, so the minimum class size is 10.