A-level · Hypothesis testing

Hypothesis testing for zero correlationExam-style questions

Testing whether a population has any linear correlation by comparing the sample PMCC r against a critical value from tables — choosing one- or two-tailed hypotheses on ρ, and concluding in context.

Start practising Read the lesson

26 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
A researcher records $n = 30$ pairs of data and finds $r = 0.30$ . She tests at the $1\%$ significance level whether there is positive correlation. The critical value is $0.4226$ . State the conclusion of the test.
Level 5 · 3 marksTry it
2
A gym records, for $n = 14$ members, the number of weekly visits and the improvement in fitness score. The PMCC is $r = 0.47$ . The $5\%$ one-tail critical value for $n = 14$ is $0.4575$ .
(a) Test, at the $5\%$ level, whether there is evidence of positive correlation. (4 marks)
(b) The gym's owner concludes 'visiting more CAUSES fitness to improve'. Comment on this claim. (1 mark)
Level 5 · 5 marksTry it
3
A motoring journalist is comparing the fuel economy of petrol cars that have different sizes of engine.

For each car in a random sample of $18$ $18$ petrol cars, she records the engine size, $c$ $c$ litres, and the fuel economy, $f$ $f$ miles per gallon. Her $18$ $18$ pairs of values are plotted on the scatter diagram below.(a) Interpret the nature of the relationship between $f$ $f$ and $c$ $c$ .

(1 mark)

The journalist then works with the logarithms of her data, using

$x = \log_{10} c \qquad y = \log_{10} f$

The product moment correlation coefficient between $x$ $x$ and $y$ $y$ is $-0.955$ $- 0.955$
(b) Test whether or not there is significant evidence of a negative correlation between $x$ $x$ and $y$ $y$ .

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 $y$ $y$ on $x$ $x$ is

$y = 1.74 - 0.59x$
(c) Use the equation of the line of best fit of $y$ $y$ on $x$ $x$ to find a model for $f$ $f$ in terms of $c$ $c$ in the form

$f = ac^{k}$

where $a$ $a$ and $k$ $k$ are constants to be found. Give the value of $a$ $a$ to 3 significant figures.

(5 marks)
Level 5 · 9 marksTry it
4
A stamp dealer keeps a record of the prices paid at auction for copies of one particular rare stamp, all in the same condition. The record covers $12$ $12$ sales, each in a different year.

For each sale, the dealer notes the number of years, $t$ $t$ , after 2010 and the price paid, £ $P$ $P$ thousand. The scatter diagram below shows the dealer's record.(a) Interpret the nature of the relationship between $P$ $P$ and $t$ $t$ .

(1 mark)

The dealer calculates $y = \log_{10} P$ $y = lo g_{10} P$ for each sale.

The product moment correlation coefficient between $t$ $t$ and $y$ $y$ is $0.997$ $0.997$
(b) Test whether or not there is significant evidence of a positive correlation between $t$ $t$ and $y$ $y$ .

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 $y$ $y$ on $t$ $t$ is

$y = 0.60 + 0.050t$
(c) Use the equation of the line of best fit of $y$ $y$ on $t$ $t$ to find a model for $P$ $P$ in terms of $t$ $t$ in the form

$P = ab^{t}$

where $a$ $a$ and $b$ $b$ are constants to be found. Give the value of $a$ $a$ and the value of $b$ $b$ to 3 significant figures.

(5 marks)
Level 5 · 9 marksTry it
5
A company owns a large number of hotels. For a random sample of 15 of its hotels, the company's operations director recorded the number, $s$ , of staff employed at the hotel and the number, $k$ , of complaints received from guests at that hotel during one year. The director plans to use these figures to test for linear correlation between $s$ and $k$ .

For the 15 hotels, the value of Pearson's product-moment correlation coefficient between $s$ and $k$ is $0.671$ , correct to 3 significant figures.
(a) Using the table below, carry out the test at the 2% significance level.

(5 marks)
(b) Describe briefly two main features of a scatter diagram illustrating the values of $s$ and $k$ for these 15 hotels. You are not required to draw the diagram.

(2 marks)
(c) The director's assistant writes the following.

“Hotels with more staff get more complaints, so it must be the extra staff that cause them. The company should therefore reduce the number of staff at every hotel.”

Comment briefly on the assistant's conclusion, with reference to the context.

(2 marks)
(d) Explain, by means of a statement about a probability, what is meant by the value 0.5324 in the table below.

Critical values of Pearson's product-moment correlation coefficient
1-tail test: 5% 2.5% 1% 0.5%
2-tail test: 10% 5% 2% 1%
$n = 14$ 0.4575 0.5324 0.6120 0.6614
$n = 15$ 0.4409 0.5140 0.5923 0.6411
$n = 16$ 0.4259 0.4973 0.5742 0.6226
$n = 17$ 0.4124 0.4821 0.5577 0.6055
(2 marks)
Level 5 · 11 marksTry it

	1-tail test: 5%	2.5%	1%	0.5%
	2-tail test: 10%	5%	2%	1%
$n = 14$	0.4575	0.5324	0.6120	0.6614
$n = 15$	0.4409	0.5140	0.5923	0.6411
$n = 16$	0.4259	0.4973	0.5742	0.6226
$n = 17$	0.4124	0.4821	0.5577	0.6055