Browse Hypothesis testing

	1-tail test: 5%	2.5%	1%	0.5%
	2-tail test: 10%	5%	2%	1%
$n = 20$	0.3783	0.4438	0.5155	0.5614
$n = 21$	0.3687	0.4329	0.5034	0.5487
$n = 22$	0.3598	0.4227	0.4921	0.5368
$n = 23$	0.3515	0.4132	0.4815	0.5256

(2 marks)

●●●●●Level 511 marksStart

The mass,

X

grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

2.28\%

of loaves have mass less than

775

g and that

91.04\%

of loaves have mass between

775

g and

831

g.
(a) Find the values of

\mu

and

\sigma

25

loaves baked in the new oven has mean mass

812.5

g.
(b) Test the baker's belief at the

5\%

n

loaves from the new oven also has mean mass

812.5

g. Find the smallest value of

n

for which this sample would provide significant evidence, at the

5\%

level, that the mean mass has changed. (2 marks)

●●●●●Level 514 marksStart

A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean

500

ml and standard deviation

4

ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of

15

bottles and finds that the sample mean volume is

502.4

ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the

5\%

level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to

2

decimal places. (3 marks)
(c) Explain what is meant by the

5\%

level of significance in this context. (1 mark)
(d) The machine is serviced again. A further random sample of

15

bottles has a mean volume of

501.8

ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)

●●●●●Level 512 marksStart

A machine dispenses a liquid into bottles. The volume dispensed is normally distributed with mean

500

ml and standard deviation

4

ml.

After the machine is serviced, the manager suspects that the mean volume may have changed. She takes a random sample of

15

bottles and finds that the sample mean volume is

502.4

ml.

You may assume that the standard deviation is unchanged.
(a) Stating your hypotheses clearly, test at the

5\%

level of significance whether the mean volume dispensed has changed. (6 marks)
(b) Find the critical region for this test, giving the boundaries to

2

decimal places. (3 marks)
(c) The machine is serviced again. A further random sample of

15

bottles has a mean volume of

501.8

ml.

Explain, without further calculation, whether this sample would lead to the same conclusion. (2 marks)

●●●●●Level 511 marksStart

A mountain-rescue team is called out at random. The number of call-outs in a week is modelled by a Poisson distribution with mean

\lambda

. Records show that the probability of no call-outs in a week is

0.1827

(a) Show that

\lambda = 1.7

2

significant figures.

(2 marks)

In the rest of this question use

\lambda = 1.7

New signs are put up on the mountain paths. The team leader wants to know whether the mean rate of call-outs has changed. She will use the number of call-outs in the

5

weeks after the signs are put up.
(b) Using a

3\%

level of significance, find the critical region for this test. State your hypotheses clearly.

(4 marks)
(c) Find the actual significance level of this test.

(2 marks)

In the

5

weeks after the signs are put up, there are

3

call-outs.
(d) Comment on this finding in the light of your critical region.

(2 marks)

The team leader now believes that the signs have reduced the mean rate of call-outs. In the next

4

weeks there is

1

call-out.
(e) Test, at the

5\%

level of significance, whether or not there is evidence to support the team leader's belief. State your hypotheses clearly.

(5 marks)

●●●●●Level 515 marksStart

The mass of flour used each day by a bakery may be modelled by a normal distribution with mean

86

kg and standard deviation

7

kg.
(a) State the probability that the mass of flour used on any given day is exactly

86

kg.

(1 mark)
(b) To attract more customers, the bakery decides to change the range of bread that it bakes.

Following the decision, it was found that over a

15

-day period a total mass of

1353

kg of flour was used.

It can be assumed that the distribution of the mass of flour used each day is still normal with standard deviation

7

kg, and that the

15

-day period can be regarded as a random sample.

Investigate, at the

5\%

level of significance, whether the mean daily mass of flour used has changed.

(7 marks)
(c) The bakery's manager claims that if a different sample of

15

days had been used, the hypothesis test in part (b) would have given the same result.

Comment on the validity of this claim.

(2 marks)

●●●●●Level 510 marksStart

Hypothesis testing for zero correlation

28 questions

LessonNot started

Amara is investigating the relationship between the price of a paperback book and the number of copies of it sold each week in a large chain of bookshops.

She takes a random sample of

15

titles and records, for each title,

its price, $p$ pounds
the number of copies sold in one week, $q$

The points on Amara's scatter diagram lie close to a curve: as

p

increases

q

decreases, steeply at first and then more gently.
(a) Interpret the nature of the relationship between

q

and

p

.

(1 mark)

Amara codes her data using the formulae

x = \log_{10} p \qquad y = \log_{10} q

The product moment correlation coefficient between

x

and

y

-0.8463

The critical value for a sample of size

15

at the

5\%

level of significance is

0.4409

(b) Test whether or not there is significant evidence of a negative correlation between

x

and

y

.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the critical value used

(3 marks)

The equation of the line of best fit of

y

x

y = 3.4 - 1.6x

(c) Use the equation of the line of best fit of

y

x

to find a model for

q

in terms of

p

in the form

q = ap^{k}

where

a

and

k

are constants to be found. Give the value of

a

to 3 significant figures.

(5 marks)

●●●●●Level 59 marksStart

Femi is investigating how the membership of a new climbing club has grown.

At the end of each of the first

10

months after the club opened he records

the time, $t$ months, since the club opened
the number of members, $m$ , of the club

The points on Femi's scatter diagram lie close to a curve: as

t

increases

m

increases, slowly at first and then more and more quickly.
(a) Interpret the nature of the relationship between

m

and

t

.

(1 mark)

Femi codes his data using the formula

y = \log_{10} m

The product moment correlation coefficient between

t

and

y

0.9317

The critical value for a sample of size

10

at the

5\%

level of significance is

0.5494

(b) Test whether or not there is significant evidence of a positive correlation between

t

and

y

.

You should

state your hypotheses clearly
use a $5\%$ level of significance
state the critical value used

(3 marks)

The equation of the line of best fit of

y

t

y = 1.62 + 0.058t

(c) Use the equation of the line of best fit of

y

t

to find a model for

m

in terms of

t

in the form

m = ab^{t}

where

a

and

b

are constants to be found. Give the value of

a

and the value of

b

to 3 significant figures.

(5 marks)

●●●●●Level 59 marksStart

Each week, the owner of a garden centre records the number,

c

, of bags of compost sold and the amount, £

a

, spent on local radio advertising during that week. The owner wants to test whether there is linear correlation between

c

and

a

.

The owner chooses a random sample of 22 weeks and finds that, for these weeks, the product-moment correlation coefficient between

c

and

a

r = 0.583

(3 sf).
(a) Using the table below, carry out the test at the 1% significance level.

(5 marks)
(b) Describe briefly two main features of a scatter diagram of the 22 pairs of values of

c

and

a

. You do not need to draw the diagram.

(2 marks)
(c) The owner makes the following statement.

“The value of

r

	1-tail test: 5%	2.5%	1%	0.5%
	2-tail test: 10%	5%	2%	1%
$n = 20$	0.3783	0.4438	0.5155	0.5614
$n = 21$	0.3687	0.4329	0.5034	0.5487
$n = 22$	0.3598	0.4227	0.4921	0.5368
$n = 23$	0.3515	0.4132	0.4815	0.5256

(2 marks)