Question

Do taller adults make more money? The authors of a paper investigated the association between height and earnings. They used the simple linear regression model to describe the relationship between x = height (in inches) and y = weekly gross earnings in dollars in a random sample of men. The paper reported that the slope of the estimated regression line was b = 0.022 and the standard deviation of b was sb = 0.005. Carry out a hypothesis test using α = 0.05 to decide if there is convincing evidence of a significant linear relationship between height and the weekly earnings. You can assume that the basic assumptions of the simple linear regression model are met. State the appropriate null and alternative hypotheses. H0: β > 0 Ha: β < 0 H0: β = 0 Ha: β > 0 H0: β = 0 Ha: β < 0 H0: β = 0 Ha: β ≠ 0 H0: β < 0 Ha: β > 0 Assume the study was based on a sample of 30 men, find the test statistic and P-value. Use a table or technology. Round your test statistic to one decimal place and your P-value to three decimal places.) t = P-value = What can you conclude? Fail to reject H0. We have convincing evidence of a significant linear relationship between height and weekly earnings. Fail to reject H0. We do not have convincing evidence of a significant linear relationship between height and weekly earnings. Reject H0. We have convincing evidence of a significant linear relationship between height and weekly earnings. Reject H0. We do not have convincing evidence of a significant linear relationship between height and weekly earnings.

Answer 1

The null and alternative hypotheses are :

against

Here,

The estimated slope b = 0.022

standard error of estimated slope SE(b) = 0.005

but sample size n = 30

Now,

The test statistic can be written as

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

We reject H₀ at 0.05 level of significance if P-value < 0.05

Now,

The value of the test statistic

P-value

Since p-value < 0.05, so we reject H₀ at 0.05 level of significance.

Conclusion : Reject H0. We have convincing evidence of a significant linear relationship between height and weekly earnings.

b t : SE(b)

Lobs=4.4

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