A study of the role of spreadsheets in planning in 55 small...

A study of the role of spreadsheets in planning in 55 small firms defined $Y=$ "satisfaction with sales growth" and $X=$ "executive commitment to planning." Analysis yielded an overall correlation of $r=.3043$. Do a two-tailed test for zero correlation at $\alpha=.025$.

Key Concepts

t-distribution Transformation

When testing the significance of a Pearson correlation coefficient, the value is transformed into a t-statistic using the formula t = r * sqrt((n-2) / (1-r^2)), where n is the sample size. This transformation accounts for the sampling variability and allows the use of the t-distribution to determine whether the correlation is statistically significant, with degrees of freedom equal to n-2.

Pearson Correlation Coefficient

The Pearson correlation coefficient measures the strength and direction of the linear relationship between two quantitative variables. It ranges from -1 to 1, where values close to 1 indicate a strong positive association, values close to -1 indicate a strong negative association, and values near 0 suggest no linear relationship.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to decide if there is enough evidence in a sample of data to infer that a particular condition is true for the entire population. In correlation testing, the null hypothesis typically states that there is no relationship between the two variables (zero correlation), while the alternative hypothesis indicates that a relationship exists.

Two-tailed Test

A two-tailed test is used when deviations from the null hypothesis can occur in either direction (either positive or negative correlations). This type of test allocates the significance level equally across both tails of the t-distribution, ensuring that extreme values on either end are considered for rejecting the null hypothesis.

Significance Level

The significance level (alpha) defines the threshold for deciding whether the observed results are due to chance. In this context, an alpha of 0.025 means that there is a 2.5% risk of falsely rejecting the null hypothesis when it is actually true. This level determines the critical t-values used to decide if the computed t-statistic from the correlation coefficient indicates a statistically significant result.

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