Download Practice Exam 1 Question 6 Do strong magnetic fields have an effect

Practice Exam 1 Question 6 Do strong magnetic fields have an effect on the early development of mice? A study performed on 120 one-­‐month-­‐old mice. Each is subjected to a strong magnetic field for one week and no magnetic field for one week, the order being decided randomly for each mouse. The weight gain in grams during each week for each mouse is recorded. The weight gains with no magnetic field are approximately normally distributed with a mean of 25.1 grams and a standard deviation of 3.1 grams, while the weight gains in the magnetic field are approximately normally distributed with a mean of 17.3 grams and a standard deviation of 5.7 grams. Of the 120 mice, 101 gained more weight with no magnetic field. (a) Calculate and interpret a 95% confidence interval for the proportion of mice that gain more weight with no magnetic field. PART 1 101
𝑝=
= 0.84
120
𝑛 = 120
𝑞 − 0.16
Part 2 Check Conditions 𝑛𝑝 = 120 . 84 = 101 > 10
𝑛𝑞 = 120 . 16 = 19 > 10
PART 3 Calculate Confidence interval Find z* : Invtnorm (.025) = 1.96 𝑝 ± 𝑧∗
𝑝𝑞
𝑛
. 84 ∗ .16
. 84 ± 1.96
120
.84 ± 0.065 (.777, .907) PART 4 I am 95% confident that the true propotion to mice that gained more weight with no magnetic field is between .777 and .901 (b) Suppose two of these mice were chosen at random. What is the probability that the first mouse gained more weight with no magnetic field than the second mouse gained with a magnetic field? WE are talking about the difference in the mean weight with and with out a magnetic field. 𝑑 = 𝑥 − 𝑦 = 25.1 − 17.3 = 7.8 𝑔
𝑠! = 𝑠!! + 𝑠!! = 3.1! + 5.7! = 6.5 If there is no difference, the value of d is 0 𝑥 − 𝜇 0 − 7.8
=
= −1.2 𝜎
6.5
P-­‐Value = NormalCDF(-­‐1.2,10) = .885 The probability that the mouse without the magnetic field gained more weight than the mouse with the magnetic field is 0.885. (c) Based on your answers to (a) and (b), is there evidence that for mice, weight gain with no magnetic field and in a magnetic field are independent? Because 0.885 lies on the interval (0.777, 0.907) there is no evidence to show weight gain with or without a magnetic field is not independent. (d) In a test for variances on whether or not the variances of the two sets of weight gains are significantly different, 𝐻! : 𝜎!! = 𝜎!! , 𝐻! : 𝜎!! ≠ 𝜎!! , with α = 0.05, for samples 𝑧=
!!
of this size, the critical cut-­‐off scores for !!! are 0.698 (left tail of 0.025) and 1.43 !
(right tail of 0.025). What is the resulting conclusion about variances? !!
!.!!
WE are testing !!! = !.!! = 3.38 (If you did the inverse its 0.296). Both of those values !
are outside the range of critical values (0.698, 1.43). Therefore they are statistically significant, and there is evidence to show that the variances of the weight gained by mice in an magnetic field is different that the variance of weight gained by mice without a magnetic field.