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Correlation

Correlation. A relationship between two variables. Explanatory (Independent) Variable. Response (Dependent) Variable. y. x. Hours of Training. Number of Accidents. Shoe Size. Height . Cigarettes smoked per day. Lung Capacity. Score on SAT. Grade Point Average. Height. IQ.

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Correlation

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  1. Correlation A relationship between two variables Explanatory (Independent) Variable Response (Dependent) Variable y x Hours of Training Number of Accidents Shoe Size Height Cigarettes smoked per day Lung Capacity Score on SAT Grade Point Average Height IQ What type of relationship exists between the two variables and is the correlation significant?

  2. Scatter Plots and Types of Correlation x = hours of training y = number of accidents 60 50 40 Accidents 30 20 10 0 0 2 4 6 8 10 12 14 16 18 20 Hours of Training Negative Correlation–as x increases, y decreases

  3. Scatter Plots and Types of Correlation x = SAT score y = GPA 4.00 3.75 3.50 3.25 GPA 3.00 2.75 2.50 2.25 2.00 1.75 1.50 300 350 400 450 500 550 600 650 700 750 800 Math SAT Positive Correlation–as x increases, y increases

  4. Scatter Plots and Types of Correlation x = height y = IQ 160 150 140 130 IQ 120 110 100 90 80 60 64 68 72 76 80 Height No linear correlation

  5. Correlation Coefficient 1 –1 0 A measure of the strength and direction of a linear relationship between two variables The range of r is from –1 to 1. If r is close to –1 there is a strong negative correlation. If r is close to 1 there is a strong positive correlation. If r is close to 0 there is no linear correlation.

  6. Application Final Grade Absences x y 8 78 2 92 5 90 12 58 15 43 9 74 6 81 95 90 85 80 75 Final Grade 70 65 60 55 50 45 40 0 2 4 6 8 10 12 14 16 Absences X

  7. Computation of r xy xy y2 x2 1 8 78 2 2 92 3 5 90 4 12 58 5 15 43 6 9 74 7 6 81 624 184 450 696 645 666 486 64 4 25 144 225 81 36 6084 8464 8100 3364 1849 5476 6561 57 516 3751 579 39898

  8. Hypothesis Test for Significance r is the correlation coefficient for the sample. The correlation coefficient for the population is (rho). For a two tail test for significance: (The correlation is not significant) (The correlation is significant) For left tail and right tail to test negative or positive significance: The sampling distribution for r is a t-distribution with n – 2 d.f. Standardized test statistic

  9. Test of Significance You found the correlation between the number of times absent and a final grade r = –0.975. There were seven pairs of data.Test the significance of this correlation. Use = 0.01. 1. Write the null and alternative hypothesis. (The correlation is not significant) (The correlation is significant) 2. State the level of significance. = 0.01 3. Identify the sampling distribution. A t-distribution with 5 degrees of freedom

  10. 4.032 –4.032 Rejection Regions Critical Values ± t0 t 0 4. Find the critical value. 5. Find the rejection region. 6. Find the test statistic.

  11. t 0 –4.032 –4.032 7. Make your decision. t = –9.811 falls in the rejection region. Reject the null hypothesis. 8. Interpret your decision. There is a significant correlation between the number of times absent and final grades.

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