{"id":3783,"date":"2013-12-12T17:56:43","date_gmt":"2013-12-12T22:56:43","guid":{"rendered":"http:\/\/www.thejuliagroup.com\/blog\/?p=3783"},"modified":"2013-12-12T19:44:07","modified_gmt":"2013-12-13T00:44:07","slug":"maria-and-eric-meet-z-scores","status":"publish","type":"post","link":"https:\/\/www.thejuliagroup.com\/blog\/maria-and-eric-meet-z-scores\/","title":{"rendered":"Maria and Eric meet z-scores"},"content":{"rendered":"

\"Maria<\/a><\/p>\n

One of the problems many students have when first learning statistics is deciding when to reject the null hypothesis. Z is small and low probability means it is not likely to occur so you reject, right? (Wrong!). P > .86 and when you have a large z-score you reject the null hypothesis, so with p = .86 you reject, right? (Wrong!)<\/p>\n

Enter Maria and Eric to help us explain z-scores. Eric is 6 foot 4, or 76 inches tall. That is a high number, both in mathematical terms and off of the ground. I want to determine if Eric’s height is significantly different from the mean. I use the heart data set included with the SAS Web Editor to compute mean and standard deviation for an adult male, as so:<\/p>\n

proc sort data=sashelp.heart out=temp ;
\nby sex ;
\nproc univariate data=temp;
\nvar height \u00a0;
\nby sex ;<\/p>\n

I find that the mean is 67.6 and the standard deviation is 2.7. I then compute my z-score which is the obtained value \u00a0of 76 inches, minus the mean value of 67.6 divided by the standard deviation of 2.7. This gives me a z-score of 3.1 which tells me that Eric is 3.1 standard deviations above the mean.<\/p>\n

Listen carefully here — there is a SMALL probability of LARGE differences from the mean.<\/p>\n

A z-score of 1.96 occurs less than 5% of the time, that is about two standard deviations from the mean. How often does a z-score of 3.1 occur? p < .002. So, even though he is a LARGE difference from the average height, \u00a0people who are that tall represent a small proportion of the population.<\/p>\n

We would therefore REJECT that null hypothesis that there is NO difference between Eric’s height and the average and conclude that he is significantly taller than average.<\/p>\n

Since Maria just sniffed disrespectfully,<\/p>\n

I could have told you that!<\/p><\/blockquote>\n

(I can hear you over the Internet) … we will now examine Maria.<\/p>\n

She is 5 foot 4, or 64 inches. The average height for a woman is 62.6 inches and the standard deviation is 2.5. \u00a0Her z-score is (64-62.6)\/ 2.5 = .56 and the probability of a z-score that high or larger is almost 60%, \u00a0p> .59 . So, she differs a SMALL amount from the average and that will happen a LARGE proportion of the time.<\/p>\n

SO … would you accept or reject the null hypothesis that Maria is no different than the average height for women? Discuss.<\/p>\n

—————— Buy our game! It’s awesome.<\/p>\n

Want your children to be good at math? Want to improve your own math skills while killing animated buffalo and running around in a virtual world? Have $9.99 ? It’s your lucky day. Click here to buy Spirit Lake: The Game<\/a><\/p>\n

\"Splash<\/a><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

One of the problems many students have when first learning statistics is deciding when to reject the null hypothesis. Z is small and low probability means it is not likely to occur so you reject, right? (Wrong!). P > .86 and when you have a large z-score you reject the null hypothesis, so with p…<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[11],"tags":[],"class_list":["post-3783","post","type-post","status-publish","format-standard","hentry","category-statistics"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts\/3783","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/comments?post=3783"}],"version-history":[{"count":8,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts\/3783\/revisions"}],"predecessor-version":[{"id":3794,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/posts\/3783\/revisions\/3794"}],"wp:attachment":[{"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/media?parent=3783"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/categories?post=3783"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.thejuliagroup.com\/blog\/wp-json\/wp\/v2\/tags?post=3783"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}