Question

The number of hours 10 students spent studying for a test and their scores on that test are shown in the table. Is there enough evidence to conclude that there is a significant linear correlation between the​ data? Use alphaequals0.05.

Hours, x   Test score, y
0   40
1   40
2   55
4   49
4   65
5   67
5   73
6   70
7   81
8   93

Setup the hypothesis for the test.

Upper H 0H0​:

rhoρ

▼

less than or equals≤

less than<

equals=

greater than>

greater than or equals≥

not equals≠

0

Upper H Subscript aHa​:

rhoρ

▼

less than or equals≤

less than<

equals=

greater than>

greater than or equals≥

not equals≠

0

Identify the critical​ value(s). Select the correct choice below and fill in any answer boxes within your choice.

​(Round to three decimal places as​ needed.)

A.The critical value is

nothing.

B.The critical values are

minus−t 0t0equals=nothing

and

t 0t0equals=nothing.

Calculate the test statistic.

tequals=nothing

​(Round to three decimal places as​ needed.)

What is your​ conclusion?

There

▼

is not

is

enough evidence at the

55​%

level of significance to conclude that there

▼

is

is not

a significant linear correlation between hours spent studying and test score.

Answer 1

Let denotes the true correlation coefficient between hours and test score.

To test against

Here

sample correlation coefficient

and sample size

The test statistic can be written as

which under H₀ follows a t distribution with n-2 df.

We reject H₀ at 5% significance level if

Now,

The value of the test statistic =

and critical value

Since , so we reject H₀ at 5% significance level.

There is enough evidence at the 5​% level of significance to conclude that there is a significant linear correlation between hours spent studying and test score.