This study examines the relationship between student scores on a measure of anxiety and the number of hours each student spent studying. The expectation is that students who are more anxious will spend more time studying. The best statistic for this study is correlation, since the correlation indicates whether the two variables change together, and if they do, how closely. Pearson’s r, the typical correlation statistic, is a number between -1 and 1, with -1 a perfect negative correlation, 0 no correlation and 1 meaning a perfect positive correlation. The effect size can be obtained by finding r2(Howell, 2013)

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When using statistics, it is important for the researcher to know what kind of relationship he or she is anticipating. Based on this expectation, the researcher will create a null hypothesis and an alternate hypothesis (Jakobsen et al., 2014). The alternate hypothesis expresses the expected effect, while the null hypothesis expresses lack of effect. In this study, the null hypothesis is that there is no correlation between student anxiety scores and number of study hours; the alternate hypothesis is that there is a correlation between the 2 variables.

H0: r = 0
H1: r ≠ 0

The obtained correlation (r) for this data is 0.565. This is a moderate correlation. The effect size (r2) is 0.3196, meaning that 31.96% of the variation in Y (number of study hours) can be explained by x (student anxiety score). Since two-thirds of the variance is still unexplained, there must be other factors influencing the number of hours a student studies (Howell, 2013).

Correlation coefficients and effect size are not enough to support an alternate hypothesis; the researcher must make allowance for errors and variance. This is the purpose of testing for significance. A test for significance essentially tells whether the obtained sample value is far enough away from an estimated value so that the null hypothesis can be rejected. In this case, it is necessary to determine whether or not r is far enough away from 0 so that it is unlikely to have occurred by chance. An alpha level of 0.05 is a standard value (Howell, 2013).

A table of critical values is used to determine whether a correlation coefficient is statistically significant (Jakobsen et al., 2014). The tables are based on two numbers: the alpha level and the degrees of freedom (typically N-2). For this correlation, the critical value is 0.632. The obtained correlation here was 0.565. As a result, the null hypothesis cannot be rejected. It is possible that repeating the study with a larger sample would be able to find a statistically significant correlation, since this sample was very small (Schneider, 2013).

There are two major types of statistical errors: Type 1 and Type 2. A Type 1 error is rejecting the null hypothesis when it should not be rejected; a Type 2 error is failing to reject the null hypothesis when it should be rejected. When a researcher chooses an alpha level, he or she is basically choosing what percentage of Type 1 errors would be acceptable. A alpha level of 0.05 means that 5% of the time a null hypothesis will be rejected when it should not be. This is also called a false positive. In most scientific research, Type 1 errors (finding an effect when there is none) is considered more of an issue than Type 2 errors (Howell, 2013).

For this data, one can conduct a t-test (because the sample size is small) on the correlation coefficient itself to determine significance.

t = (r√N-2)/(√1-r2)
t = 1.91, critical t = 2.306

According to this calculation, the null hypothesis cannot be rejected — the obtained t is less than the critical t (Schneider, 2013). As mentioned above, repeating the study with a sample size of at least 30 might make it more likely that a significant correlation could be determined.

It would also be possible to conduct a one-sample t-test on this data, if the population mean could be estimated. Assumptions would include

— random sampling
— defined population
— interval or ratio scale measures (number of study hours is ratio; anxiety score is ordinal)
— normally distributed population (Schneider, 2013).

The t-test would be conducted in order to determine if the mean number of hours spent studying by the population as a whole was essentially the same as that of the sample. However, this test would say nothing about the significance of the correlation between anxiety scores and hours studying (Jakobsen et al., 2014).

In order to calculate ANOVA, three or more groups or levels are needed. For this study, the students might be placed in three levels based on their anxiety scores:

Group 1 — scores of 1-3 (n=3)
Group 2 — scores of 4-5 (n=3)
Group 3 — scores of 6-12 (n=4) (with ANOVA, n does not have to be the same for each group)

The mean and standard deviation for the number of study hours by members of each group would then be calculated. The null hypothesis would be that the mean number of hours spent studying is equal across the groups. The alternate hypothesis would be that the mean number of study hours is different for at least one of the groups. The F statistic (based on the F distribution) is calculated to test for significance of ANOVA conclusions (Jakobsen et al., 2014).

Use of ANOVA does involve certain assumptions which are probably not met here due to the small sample size:

— normally distributed population
— independent samples
— equal variances of the populations.

In addition, ANOVA is technically restricted to interval/ratio data, but it is often used with ordinal data nonetheless (Howell, 2013).

References
• Howell, D. (2013). Fundamental statistics for the behavioral sciences. Cengage Learning.
• Jakobsen, J. C., Gluud, C., Winkel, P., Lange, T., & Wetterslev, J. (2014). The thresholds for statistical and clinical significance-a five-step procedure for evaluation of intervention effects in randomised clinical trials. BMC medical research methodology, 14(1), 34.
• Schneider, J. W. (2013). < i> Caveats for using statistical significance tests in research assessments. Journal of Informetrics, 7(1), 50-62.