 # MATH 101 Quantitative Reasoning

## Question:

This is an assignment that is solely for the individual and collaboration is not possible.

Refer to all sources.

The Pearson linear correlation test is shown below.

Please describe the purpose and assumptions for the test.

Please explain the meanings of the following output items.

1) Statistics t

2) Degrees Of Freedom df

4) Alternative hypothesis

5) 95% confidence range

Pearson linear correlation is statistics which measures the degree of linear relationship between two variables, such as and (Hassett-Stewart 2006).

It is also known as Pearson’s correlation coefficient.

It can be between -1 to +1.

Close to indicate perfect relationships between the associated variables.

A negative value means that there is a negative correlation between the variables. Positive values, however, indicate a positive relationship. (Jackson (2015)

Sharma (2005) explains that Pearson’s correlation rests on the following assumptions.

There are many independent forces that affect the variables, so they produce a normal distribution.

The linear association exists between the two variables.

The scatter diagram’s plots will show that the linear association between the variables is evident.

There is a cause/effect association between the forces that influence the distributions of items in each series.

What Does Output Mean?

It is a standard value calculated from the sample data in a hypothetical distribution, particularly when other parameters such as standard deviation are unknown. (Richardson (2011)

It is used in hypothesis tests to compare the mean value of two variables.

To make a decision in a hypothesis test, the t-statistic is compared to its critical value. It is set at a certain significance level and degree.

If tstatistic is higher than the critical value it means that there is a significant difference between variables (Montgomery & Runger 2010).

Freedom of movement

It is combined with the significance to determine the critical values of statistical tests from respective tables like F, t and Chi-square.

Value is the probability that you will find values equal or greater to the observed results (Brunson, 1987).

If the value is higher, it means that there is statistical significance. Conversely, a lower value indicates that there is no statistical significance.

Large p values indicate that the data are in agreement with the null hypothesis. (Ruppert (2014)

For hypothesis tests, the cut-off point is – value. This is used to determine statistical significance.

Null hypothesis is accepted if – value is less than observed results.

This is a case where alternative hypotheses are rejected. It means that statistical significance exists between variables.

Alternative hypothesis: This hypothetical statement is in opposition to the null hypothesis.

It is a hypothesis which requires support evidence (Crossley, 2000).

It’s usually used to indicate statistical significance between two or three variables.

95% confidence interval: The confidence interval is an estimate of the interval based on observed results. It has a 95% confidence level. Sim&Wright, 2000.

The confidence interval, which is used in most cases to determine a parameter’s range, or a minimum and maximal value within a specified level of significance (in our case 95%), is used.

In hypothesis tests, the 95% confidence range is sometimes used, particularly when one needs to determine if a given value (say, mean) is within a specified range.

Sample estimate: This is the estimated value of the population parameter ors from the data. It is usually calculated from the sample mean. (Traat (2013).

For large population parameters that are difficult to determine by direct means, sample estimates are used for data analysis. Therefore, a smaller sample and one that is easier to analyze is chosen.

Pearson’s correlation coefficient between X, Y is 0. This indicates that they have a negative linear relation.

The correlation coefficient between X, Y, and is also smaller than expected and is closer to 0 then it should be, which suggests that there is a low degree of relationship between them (Francis 2004,).

Hypothesis Testing

Can zero correlation of (?) be rejected at the 0.05 level of significance?

Hypotheses

Pearson Correlation: Results

The critical value of the hypothesis must be determined from the -table using a 95% significance threshold and 18 degrees of freedom in order to make a decision.

Results and discussion

The decision will be made based on the comparison between two statistics: the computed with its critical value and the computed with significance level 0.05 (Weakliem 2016).

If -computed is higher than the critical worth of, null hypothesis will be rejected.

Goos & Meintrup (2016, p. 3) state that an alternative hypothesis will prevail in this situation. This indicates statistical significance between variables.

The acceptance of the alternative hypothesis when -computed exceeds the critical value null hypothesis indicates that there is no statistical significaiton between variables.

Conversely, when the value is less than 0.05, statistical significance is between variables. Thus, null hypothesis is rejected. Instead, alternative hypothesis is accepted (Rupert (2014)).

Null hypothesis is accepted if it is less than 0.05. This means that there’s no significance between variables.

Based on the above investigation, the – computed (-0.455) is lower than the critical values of, 1.734.

The same applies to a value greater than 0.05.

This indicates that the statistical significance of the variables is negligible and null hypothesis will therefore be accepted.

This indicates that the Pearson’s correlation of and is null at the 5% level.

This implies that X, Y and Y are not related.

This revelation is false due to the fact the sample estimate for correlation coefficient is not null.

Sample correlation is, which shows that there is a negative linear relationship between X & Y.

This indicates that at 5% significance, sample data is insufficient to make statistically significant decisions about the correlation between Z and Y.

Confidence Interval

95% confidence interval.

This shows that Pearson’s correlation between and is between and.

This is statistically true since the correlation sample estimate falls within this range.

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