The correlation coefficient r is a sample statistic. what does it tell us about

Question

The correlation coefficient r is a sample statistic. what does it tell us about the value of the population correlation coefficient (Greek letter rho)? You do not know how to build the formal structure of hypothesis tests of yet. However, there is a quick way to determine if the sample evidence based on is strong enough to conclude that there is some population correlation between the variables. In other words, we can use the value of r to determine if 0. we do this by comparing the value Irl to an entry in the correlation table. The value of in the table gives us the probability of concluding that 0 when, in fact, = 0 and there is no population correlation. We have two choices for : = 0.05 or = 0.01 Critical Values for Correlation Coefficient r =0.01 1.00 0.99 0.96 0.92 0.87 0.83 0.80 0.76 0.73 0.71 =0.05 =0.01 0.68 0.66 0.64 0.61 0.61 0.59 0.58 0.56 0.55 0.54 =0.05 =0.01 0.53 0.52 0.51 0.50 0.49 0.48 0.47 0.46 =0.05 23 0.41 24 0.40 25 0.40 26 0.39 27 0.38 28 0.37 29 0.37 30 0.36 3 1.00 4 0.95 5 0.88 6 0.81 7 0.75 8 0.71 0.67 10 0.63 11 0.60 2 0.58 13 0.53 14 0.53 15 0.51 16 0.50 0.48 18 0.47 19 0.46 20 044 043 22 0.42 17 (a) Look at the data below regarding the variables x = age of a Shetland pony and y = weight of that pony. Is the value of Irl large enough to conclude that weight and age of Shetland ponies are correlated? Use = 0.05. (Use 3 decimal places.) 18 12 60 95 140 184 170 3 6 20 r0.9724 critical r 0.959 Conclusion Reject the null hypothesis, there is sufficient evidence to show that age and weight of Shetland ponies are correlated Reject the null hypothesis, there is insufficient evidence to show that age and weight of Shetland ponies are correlated Fail to reject the null hypothesis, there is insufficient evidence to show that age and weight of Shetland ponies are correlated

Explanation / Answer

a)

For the 1st dataset there are 5 pair of (X,Y). so n=5. The point is that you don't know how to see the critical values from the table. Refer to n=5 row of the table. As we are instructed to to use alpha=0.05, then refer to column alpha=0.05 of the n=5 row. you find that critical value is 0.88

So critical r = 0.88

As |r| = 0.9724 > 0.88= critical value then we will null hypothesis and there is sufficient evidence to show that age and weight of shetland ponies are correlated.

Option (a) is true.

b) In R write the following code to the value of r.

Code:

> x<-c(1004,975,992,935,978,938)
> y<-c(40,100,65,145,75,154)
> cor(x,y)

Output:

[1] -0.984398

So the value of r is -0.9844 , I think you have reported the absolute value of r and got the wrong answer. But they are not asking for the absolute value. So answer will be r = -0.9844

Here in the 2nd dataset there are 6 pair of observations.

Similarly as above refer to n=6 row of the table. As we are instructed to to use alpha=0.01, then refer to column alpha=0.01 of the n=6 row. you find that critical value is 0.92.

So critical r = 0.92

As |r| = absolute value of r = 0.9844 > critical value = 0.92 then we will reject null hypothesis there is sufficient evidence to show that barometric pressure and maximum wind speed of cyclone are correlated.

Option (a) is true.

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