A study measures the dimensions of adult penguin bills for three species (Adelie. Gentoo and Chinstrap) living on an island. Bill lengths and bill depths (millimetres) were taken. as shown in the following diagram: I Bill length IBill depth Question 1 In the study it is believed that there is a negative linear relationship between bill lengths and bill depths for the penguins Define p as the population correlation coefficient between bill length and bill depth. The appropriate null and alternative hypotheses to test are Ho:p=0vsH1:p0 Hozp=0vslep0VSH11p0 Question 2 Measurements are taken from a random selection of 12 males (4 from each species) and 12 females (4 from each species). Based on the data, the test statistic for the hypothesis test in Question 1 is 4.109, The degrees of freedom of the t distribution to be used to find the pvalue is 20 22 23 24 Question 3 This study concludes that there is weak evidence to suggest that there is a negative linear relationship between bill lengths and depths in penguins (p = 0076) no evidence to suggest that there is a negative linear relationship between bill lengths and depths in penguins (p = 0.140) 0 no evidence to suggest that there is a negative linear relationship between bill lengths and depths in penguins (p = 0.279) no evidence to suggest that there is a negative linear relationship between bill lengths and depths in penguins (p = 0.860) Question 4 The lower limit and the upper limit of a 95% confidence interval for population mean bill length of penguins are 43.54 mm and 47.46 mm. The margin of error is 1.34 mm 1.96 mm 2.40 mm 3.92 mm Question 5 Now suppose that you are calculating a 90% confidence interval for population mean bill length of penguins using the same sample data as in Question 4. The 90% confidence interval will be either narrower or wider than the 95% condence interval the same as the 95% confidence interval narrower than the 95% confidence interval wider than the 95% confidence interval Question 6 The researchers further believe that the mean bill lengths for the 3 species are different. Define M1 , M2 and M3 as the population mean bill lengths of the species Adelie, Gentoo and Chinstrap, respectively. The appropriate null and alternative hypotheses for the one-way ANOVA to test this belief are O Ho : M1 # M2 # M3 VS H1 : M1 = M2 = M3 O Ho : M1 = M2 = M3 VS H1 : M1 # M2 #M3 O Ho : M1 = M2 = M3 VS H1 : Not all population means are equal O Ho : M1 = M2 = M3 VS H1 : M1 > M2 # 13 Question 7 The incomplete ANOVA table below is for the hypothesis test in Question 6: Source DF Sum Sq Mean Sq F value Pr(>F) Species 358.9 Residuals 136.9 The degrees of freedom for Species and Residuals respectively are O 1,23 O 2,21 O 2,23 O 3,24 Question 8 Based on the ANOVA table in Question 7, penguin species explains O 27.6% of variability in bill lengths 32.4% of variability in bill lengths 53.0% of variability in bill lengths O 72.4% of variability in bill lengths