a) The following table shows, for six commercial banks, the percentage return on equity in 2020, and the level of executive bonus payments expressed as a percentage of total payroll: x = % return on equity, 2020 3.5 4.7 2.1 6.0 1.4 2.1 y = bonus payments as % of payroll, 2020 6.2 2.9 4.4 1.7 5.2 4.8 The sample means and sample variances for x and y are as follows x̄= 3.3. ȳ= 4.2 sx 2 = 3.156 sy 2 = 2.668 (i) Calculate the sample correlation coefficient, r. (ii) What does the sample correlation coefficient suggest about the relationship between the percentage return on equity and the level of executive bonus payments? (iii) Test the null hypothesis that the population correlation is zero against a two-sided alternative hypothesis, using a significance level of 0.05. (iv) What does the outcome of your hypothesis test in part (iii) suggest about the relationship between the percentage return on equity and the level of executive bonus payments?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
a) The following table shows, for six commercial banks, the percentage return on equity in 2020, and the level of executive bonus payments expressed as a percentage of total payroll:
x = % return on equity, 2020 3.5 4.7 2.1 6.0 1.4 2.1
y = bonus payments as % of payroll, 2020 6.2 2.9 4.4 1.7 5.2 4.8
The sample means and sample variances for x and y are as follows
x̄= 3.3. ȳ= 4.2 sx 2 = 3.156 sy 2 = 2.668
(i) Calculate the sample
(ii) What does the sample correlation coefficient suggest about the relationship between the percentage return on equity and the level of executive bonus payments?
(iii) Test the null hypothesis that the population correlation is zero against a two-sided alternative hypothesis, using a significance level of 0.05.
(iv) What does the outcome of your hypothesis test in part (iii) suggest about the relationship between the percentage return on equity and the level of executive bonus payments?
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