6 A student wonders if there is any correlation between download and upload speeds of data to and from the internet. The student decides to carry out a hypothesis test to investigate this and so measures the download speed \(x\) and upload speed \(y\) in suitable units on 20 randomly chosen occasions. The scatter diagram below illustrates the data which the student collected.
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- Explain why the student decides to carry out a test based on the product moment correlation coefficient.
Summary statistics for the 20 occasions are as follows.
$$\sum x = 342.10 \quad \sum y = 273.65 \quad \sum x ^ { 2 } = 5989.53 \quad \sum y ^ { 2 } = 3919.53 \quad \sum x y = 4713.62$$
- In this question you must show detailed reasoning.
Calculate the product moment correlation coefficient.
- Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is any correlation between download speed and upload speed.
- Both of the variables, download speed and upload speed, are random.
Explain why, if download speed had been a non-random variable, the student could not have carried out the hypothesis test to investigate whether there was any correlation between download speed and upload speed.