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Inferential Statistics · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A factory produces metal rods with a nominal length of

100\,\text{cm}

. The production manager suspects the machine is malfunctioning and producing rods with a mean length different from

100\,\text{cm}

. A random sample of

50

rods gives a sample mean of

\bar{x} = 99.7\,\text{cm}

. The population standard deviation is known to be

\sigma = 0.5\,\text{cm}

. A hypothesis test is conducted at the

5\%

significance level.

(a)

State the null hypothesis

H_0

and the alternative hypothesis

H_1

. [1 mark]

(b)

Calculate the

p

-value for this test. [3 marks]

(c)

State the conclusion of the test, justifying your answer with reference to the

p

-value. [1 mark]

Solutions

2FoundationSAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A pharmaceutical company claims that a new drug reduces blood pressure by an average of

12\,\text{mmHg}

. A medical researcher believes the actual mean reduction is less than

12\,\text{mmHg}

. A random sample of 40 patients yields a sample mean reduction of

11.2\,\text{mmHg}

. The population standard deviation is

3.5\,\text{mmHg}

. A hypothesis test is conducted at the

1\%

significance level.

(a)

State the null and alternative hypotheses for this test. [1 mark]

(b)

Calculate the

z

-test statistic. [2 marks]

(c)

State the critical value and hence determine whether the null hypothesis should be rejected. Justify your answer. [2 marks]

Solutions

3MasterySAQ-SHypothesis testing and confidence intervalss5 marksPaper 1~8 min

A manufacturer claims that the mean lifespan of a certain type of battery is

1200

hours. A consumer group suspects the mean lifespan is less than

1200

hours. They test a random sample of

50

batteries and find a sample mean lifespan of

1175

hours. The population standard deviation is known to be

80

hours. A hypothesis test is carried out at the

5\%

significance level.

(a)

State the null hypothesis

H_0

and the alternative hypothesis

H_1

for this test. [1 mark]

(b)

Calculate the value of the test statistic

z

, where

z = \dfrac{\bar{x} - \mu}{\sigma / \sqrt{n}}

. [2 marks]

(c)

The critical value for this test is

z_c = -1.645

. Calculate the

p

-value of the test and hence determine whether the consumer group's suspicion is supported by the evidence. Justify your answer. [2 marks]

Solutions

4ChallengeSAQ-LZ-scores and t-tests9 marksPaper 1~14 min

The heights of a certain species of plant follow a normal distribution. A botanist claims the mean height of this species is

52.0\,\text{cm}

. To test this claim, the botanist measures the heights (in cm) of a random sample of 12 plants:

49.2,\ 51.8,\ 47.5,\ 53.1,\ 50.4,\ 48.9,\ 52.6,\ 50.7,\ 49.8,\ 51.2,\ 48.3,\ 50.9

(a)

Calculate the sample mean

\bar{x}

and sample standard deviation

s

for these data. [4 marks]

(b)

State the null and alternative hypotheses for a two-tailed

t

-test of the botanist's claim. [1 mark]

(c)

Calculate the value of the

t

-test statistic. [2 marks]

(d)

The critical value for this test at the

5\%

significance level is

2.201

. Determine whether the evidence supports the botanist's claim. [2 marks]

Solutions

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