Example of Confidence Interval for Variance

Example of Confidence Interval for Variance The population variance gives an indication of how to spread out a data set is. Unfortunately, it is typically impossible to know exactly what this population parameter is. To compensate for our lack of knowledge, we use a topic from inferential statistics called confidence intervals. We will see an example of how to calculate a confidence interval for a population variance.​ Confidence Interval Formula   The formula for the (1 - ÃŽ ±) confidence interval about the population variance. Is given by the following string of inequalities: [ (n - 1)s2] / B ÏÆ'2 [ (n - 1)s2] / A. Here n is the sample size, s2 is the sample variance. The number A is the point of the chi-square distribution with n -1 degrees of freedom at which exactly ÃŽ ±/2 of the area under the curve is to the left of A. In a similar way, the number B is the point of the same chi-square distribution with exactly ÃŽ ±/2of the area under the curve to the right of B. Preliminaries We begin with a data set with 10 values. This set of data values was obtained by a simple random sample: 97, 75, 124, 106, 120, 131, 94, 97,96, 102 Some exploratory data analysis would be needed to show that there are no outliers. By constructing a stem and leaf plot we see that this data is likely from a distribution that is approximately normally distributed. This means that we can proceed with finding a 95% confidence interval for the population variance. Sample Variance We need to estimate the population variance with the sample variance, denoted by s2. So we begin by calculating this statistic. Essentially we are averaging the sum of the squared deviations from the mean. However, rather than dividing this sum by n we divide it by n - 1. We find that the sample mean is 104.2. Using this, we have the sum of squared deviations from the mean given by: (97 – 104.2)2 (75 – 104.3)2 . . . (96 – 104.2)2 (102 – 104.2)2 2495.6 We divide this sum by 10 – 1 9 to obtain a sample variance of 277. Chi-Square Distribution We now turn to our chi-square distribution. Since we have 10 data values, we have 9 degrees of freedom. Since we want the middle 95% of our distribution, we need 2.5% in each of the two tails. We consult a chi-square table or software and see that the table values of 2.7004 and 19.023 enclose 95% of the distribution’s area. These numbers are A and B, respectively. We now have everything that we need, and we are ready to assemble our confidence interval. The formula for the left endpoint is [ (n - 1)s2] / B. This means that our left endpoint is: (9 x 277)/19.023 133 The right endpoint is found by replacing B with A: (9 x 277)/2.7004 923 And so we are 95% confident that the population variance lies between 133 and 923. Population Standard Deviation Of course, since the standard deviation is the square root of the variance, this method could be used to construct a confidence interval for the population standard deviation. All that we would need to do is to take square roots of the endpoints. The result would be a 95% confidence interval for the standard deviation.

Dark Quotes About Life from Oscar Winner American Beauty

Dark Quotes About Life from Oscar Winner 'American Beauty' In the cult film American Beauty, no one is who they seem. The movie touches upon dysfunctional families, which may look picture-perfect on the outside.  Ã¢â‚¬â€¹American Beauty is a dark satire- a profoundly complex, yet brilliantly executed movie that tells you a tale you didnt want to hear.  It is a thought-provoking, incisive movie with a message that each family is special in its own way.​ Review the main characters and read a compilation of some of the best lines from this classic 90s movie. The Main Characters Lester Burnham  is a middle-aged non-achiever- a father with a boring career. His deadpan voiceover as the narrator communicates his lack of enthusiasm. For him, life is nothing but a series of repetitive, boring menial tasks. Beneath his patient and subservient exterior lies a frustrated man, hungry for love and recognition.Carolyn Burnham is  Lester Burnhams obnoxious and overbearing wife. Carolyn is also a character with many colors. She obsesses over every detail, often overriding her familys views to suit her whims. She is driven and capable of going to extremes to achieve her ends, but beneath her abrasive surface lies a person who is frustrated, fearful and sexually bored.  Jane Burnham,  the teenage daughter of Lester and Carolyn, suffers from an inferiority complex. She is frustrated, insecure, and depressed. She is ashamed of her parents  and is disturbed by their incessant quarrels and selfish pursuits. Jane can’t stand it when her dad lusts after her best friend Angela.Angela Hayes, who visits the Burnham family, is not without flaws. Beneath her gorgeous looks and mellifluous voice, she is a narcissist, obsessed with her looks and her desirability quotient with men. She has no qualms about the fact that her best friends father is lusting for her, and she seduces him further. Her overtly sexual aura is a facade to hide the deeply cracked insecure disaster within. Lester Quotes My name is Lester Burnham. This is my neighborhood; this is my street; this is my life. I am 42 years old; in less than a year I will be dead.   Our marriage is just for show. A commercial for how normal we are when were anything but. Both my wife and daughter think Im this gigantic loser. I didnt lose my job. Its not like Whoops! Where did my job go? I quit. Remember those posters that said, Today is the first day of the rest of your life? Well, thats true with every day except one: the day that you die. I feel like Ive been in a coma for the past twenty years. And Im just now waking up. This isnt life, its just stuff. And its become more important to you than living. Well, honey, thats just nuts. Carolyn Quotes Dont you mess with me, mister, or Ill divorce you so fast itll make your head spin! There happens to be a lot about me that you dont know, Mr. Smarty Man. Theres plenty of joy in my life. Well, I see youre smoking pot now. I think using psychotropic drugs is a very positive example to set for our daughter. Oh, you dont complain? Then I must be psychotic, then! What is this? Yeah, lets bring in the laugh-meter and see how loud it gets. Jane Quotes And well always be freaks and well never be like other people, and youll never be a freak because youre just too... perfect! I know you think my dads harmless, but youre wrong. Angela Quotes I dont think that theres anything worse than being ordinary. Everything thats meant to happen does. You want to have, like, ten thousand of his babies. Quotes by Other Characters My dad thinks I paid for all this with catering jobs. Never underestimate the power of denial. – Rocky Fitts Shes not your friend. Shes just someone you use to feel better about yourself. – Ricky Fitts But it helps me remember... I need to remember... Sometimes theres so much beauty in the world, I feel like I cant take it, and my heart is just going to cave in. – Ricky Fitts In order to be successful, one must project an image of success at all times. – Buddy Kane