Question

1. In an area of the Midwest, records were kept on the relationship between the rainfall (in inches)
and the yield of wheat (bushels per acre).

Rain (inches)	10.5	8.8	13.4	12.5	18.8	10.3	7.0	15.6	16.0
Yield (bushels/acre)	50.5	46.2	58.8	59.0	82.4	49.2	31.9	76.0	78.8

Find the correlation coefficient for the given data.

a. 4.267

b. 0.9808

c. 0.9620

d. 3.158

2.

A study was conducted to compare the average time spent in the lab each week versus course grade for computer programming students. The results are recorded in the table below:
Hours spent in lab .. Grade (percent)
10 96
11 51
16 62
9 58
7 89
15 81
16 46
10 51

Find the value of the linear correlation coefficient r.

a. 0.462

b. -0.335

c. 0.017

d. -0.284

3. Applicants for a particular job, which involves extensive travel in Spanish speaking countries,
must take a proficiency test in Spanish. The sample data below were obtained in a study of the
relationship between the numbers of years applicants have studied Spanish and their score on the test.

Number of years (x)	Score (y)
3	57
4	78
4	72
2	58
5	89
3	63
4	73
5	84
3	75
2	48

Determine whether the data provide sufficient evidence to conclude that number of years of study
and test score are positively linearly correlated. Test at the 5% significance level.

a. Test statistic: t = 0.83654
Do not reject H₀. At the 5% significance level, the data do not provide sufficient evidence to conclude that number of years of study and test score are positively linearly correlated.

b.Test statistic: t = 0.91463
Do not reject H₀. At the 5% significance level, the data do not provide sufficient evidence to conclude that number of years of study and test score are positively linearly correlated.

c.Test statistic: t = 5.3294
Reject H₀. At the 5% significance level, the data provide sufficient evidence to conclude that number of years of study and test score are positively linearly correlated.

d. Test statistic: t = 6.2525
Reject H₀. At the 5% significance level, the data provide sufficient evidence to conclude that number of years of study and test score are positively linearly correlated.

4.Given the linear correlation coefficient r and the sample size n, determine the critical
values of r and use your finding to state whether or not the given r represents a
significant linear correlation.   Use a significance level of 0.05.

r = 0.539, n = 25

a. Critical values: r = 0.396, no significant linear correlation

b. Critical values: r = 0.487, no significant linear correlation

c. Critical values: r = 0.487, significant linear correlation

d. Critical values: r = 0.396, significant linear correlation

Answer 1

1. Correlation coefficient

The formula of correlation coefficient

n = number of pairs = 9

X: Rain and Y: Yield

Correlation coefficient (r) = 0.9808

Option b is correct.

2.

The formula of correlation coefficient

n = number of pairs = 8

X: Hours spent  and Y: Percent

Correlation coefficient (r) = -0.335

Option b is correct.

3.

The formula of correlation coefficient

n = number of pairs = 10

X: Number of years and Y: Score

The null and alternative hypothesis

H₀: There is no linear correlation between the number of years of study and test scores.

H₁: There is positive linear correlation between the number of years of study and test scores.

The test statistics is

t critical value using the n - 2 that is 8 degrees of freedom with the area in one tailed as alpha 0.05 is 1.86

If critical value > test statistics then fail to reject the H0 otherwise reject the H0

Critical value 1.86 is less than test statistics that is 6.2525 so reject the null hypothesis.

Option d is correct.

d: Test statistic: t = 6.2525
Reject H₀. At the 5% significance level, the data provide sufficient evidence to conclude that number of years of study and test score are positively linearly correlated.

4.

The null and alternative hypothesis are,

H₀: There is no significant linear correlation.

H₁: There is significant linear correlation.

n = 25 , alpha = 0.05

Degrees of freedom = n - 2 = 23, the test is two-tailed test.

The critical value using the critical value table for degrees of freedom 23 and alpha 0.05 in two tailed is 0.396

Critical value = 0.396, r = 0.539

Critical value is not more than r so reject the null hypothesis.

Reject the null means there is significant linear correlation.

Option d is correct.

d. Critical values: r = 0.396, significant linear correlation.