Exercises for Question Type 8: Performing a hypothesis test for correlation under bivariate normality

Question 1

Skill question

A sample of 10 pairs of data results in a Pearson product-moment correlation coefficient of $r = 0.936$ .

At the 5% significance level, test the null hypothesis $H_{0} : ρ = 0$ against the alternative hypothesis $H_{1} : ρ > 0$ .

[6]

Question 2

Skill question

A sample of size $n = 10$ has a product-moment correlation coefficient of $r = 0.936$ .

Test at the $5%$ significance level (two‐tailed) the null hypothesis $H_{0} : ρ = 0$ against $H_{1} : ρ \neq = 0$ for this data.

[5]

Question 3

Skill question

A sample of 10 observations is taken from a bivariate normal distribution, yielding a product-moment correlation coefficient of $r = 0.936$ .

Use Fisher's $z$ -transformation to test, at the 5% significance level, the null hypothesis $H_{0} : ρ = 0.3$ against the alternative hypothesis $H_{1} : ρ \neq = 0.3$ .

[6]

Question 4

Skill question

A researcher calculated the Pearson's product-moment correlation coefficient for a sample of $n = 10$ pairs of data and obtained a value of $r = 0.936$ .

At the $1%$ significance level (one-tailed), test the null hypothesis $H_{0} : ρ = 0$ against the alternative hypothesis $H_{1} : ρ > 0$ .

[6]

Question 5

Skill question

A sample of $n = 10$ observations is taken and the product-moment correlation coefficient is calculated as $r = 0.936$ .

Test at the $1%$ significance level (two-tailed) the null hypothesis $H_{0} : ρ = 0$ against $H_{1} : ρ \neq = 0$ for this data.

[6]

Question 6

Skill question

A sample of size $n = 10$ is taken from a bivariate normal population. The sample product-moment correlation coefficient is calculated as $r = 0.936$ .

At the $1%$ significance level, test $H_{0} : ρ = 0.6$ versus $H_{1} : ρ < 0.6$ using Fisher’s $z$ -transformation.

[6]

Question 7

Skill question

Specification

Calculate the sample correlation coefficient $r$ for a given bivariate data set.

Calculate the sample correlation coefficient $r$ for the following data set:

$(345, 453), (234, 112), (45, 887), (216, 593), (543, 953), (143, 438), (576, 299), (396, 548), (222, 671), (341, 554)$

[3]

Question 8

Skill question

For a data set where $n = 10$ and the Pearson product-moment correlation coefficient is $r = 0.936$ :

At the $10%$ significance level (one-tailed), test the hypothesis $H_{0} : ρ = 0$ against $H_{1} : ρ > 0$ for this data set.

[5]

Question 9

Skill question

A study of 10 pairs of observations yielded a Pearson product-moment correlation coefficient of $r = 0.936$ .

Test at the $5%$ significance level (two-tailed) the null hypothesis $H_{0} : ρ = 0$ against the alternative hypothesis $H_{1} : ρ \neq = 0$ .

[6]

Question 10

Skill question

A researcher collects a sample of $n = 10$ pairs of observations and calculates a sample product-moment correlation coefficient of $r = 0.936$ .

At the $5%$ significance level, test the null hypothesis $H_{0} : ρ = 0.5$ against the alternative hypothesis $H_{1} : ρ \neq = 0.5$ using Fisher's $z$ transformation.

[7]

Question 11

Skill question

The following data refers to a study of the relationship between two variables, $X$ and $Y$ , where a sample size of $n = 10$ was used and the Pearson correlation coefficient was calculated as $r \approx 0.936$ .

At the $5%$ significance level (one-tailed), test $H_{0} : ρ = 0$ against $H_{1} : ρ < 0$ for this data.

[4]

Question 12

Skill question

A sample of size $n = 10$ is taken from a bivariate normal distribution, and the product-moment correlation coefficient is calculated as $r = 0.936$ .

Use Fisher's $z$ -transformation to test at the 5% level $H_{0} : ρ = 0.4$ versus $H_{1} : ρ > 0.4$ for this data.

[5]

IB Mathematics Applications & Interpretation Question Type 8: Performing a Hypothesis Test for Correlation Under Bivariate Normality Exercises

Specification