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Performing the hypothesis test

    Setting up the hypotheses:

  • Null Hypothesis: H o : ρ = 0
  • Alternate Hypothesis: H a : ρ ≠ 0

    What the hypotheses mean in words:

  • Null Hypothesis H o : The population correlation coefficient IS NOT significantly different from 0. There IS NOT a significant linear relationship(correlation) between x and y in the population.
  • Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM 0. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

    Drawing a conclusion:

  • There are two methods to make the decision. Both methods are equivalent and give the same result.
  • Method 1: Using the p-value
  • Method 2: Using a table of critical values
  • In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05
  • Note: Using the p-value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

    Method 1: using a p-value to make a decision

  • The linear regression t -test LinRegTTEST on the TI-83+ or TI-84+ calculators calculates the p-value.
  • On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight " ≠ 0 "
  • The output screen shows the p-value on the line that reads "p =".
  • (Most computer statistical software can calculate the p-value.)

    If the p-value is less than the significance level (α = 0.05):

  • Decision: REJECT the null hypothesis.
  • Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from 0."

    If the p-value is not less than the significance level (α = 0.05)

  • Decision: DO NOT REJECT the null hypothesis.
  • Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from 0."

    Calculation notes:

  • You will use technology to calculate the p-value. The following describe the calculations to compute the test statistics and the p-value:
  • The p-value is calculated using a t -distribution with n-2 degrees of freedom.
  • The formula for the test statistic is t r n 2 1 r 2 . The value of the test statistic, t , is shown in the computer or calculator output along with the p-value. The test statistic t has the same sign as the correlation coefficient r .
  • The p-value is the combined area in both tails.
  • An alternative way to calculate the p-value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

    Third exam vs final exam example: p value method

  • Consider the third exam/final exam example .
  • The line of best fit is: y ^ = -173.51 + 4.83x with r = 0.6631 and there are n = 11 data points.
  • Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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