One of Mendel\u2019s famous genetics experiments yielded 580 peas, with 428 of them green and 152 yellow. a. Find a 99% confidence interval estimate of the percentage of green peas. b. Based on his theory of genetics, Mendel expected that 75% of the offspring peas would be green. Given that the percentage of offspring green peas is not 75%, do the results contradict Mendel\u2019s theory? Why or why not?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The confidence interval for the percentage of green peas is (69.09%, 78.50%)<\/strong>.<\/p>\n\n\n\n No, the results do not<\/strong> contradict Mendel\u2019s theory because the expected proportion of green peas (75%) falls within<\/strong> the 99% confidence interval (69.09% to 78.50%).<\/p>\n\n\n\n Gregor Mendel, the father of genetics, predicted that 75%<\/strong> of the offspring peas should be green based on his laws of inheritance. In this experiment, 428 out of 580 peas were green, which gives an observed proportion of:<\/p>\n\n\n\n [ To determine if this observed proportion significantly deviates from Mendel\u2019s expected proportion, we calculate a 99% confidence interval<\/strong> for the true proportion of green peas in the population.<\/p>\n\n\n\n Using the normal approximation method<\/strong>, the standard error (SE) is calculated as:<\/p>\n\n\n\n [ The Z-score for a 99% confidence level<\/strong> is 2.576<\/strong>.<\/p>\n\n\n\n The margin of error (ME)<\/strong> is:<\/p>\n\n\n\n [ Thus, the 99% confidence interval<\/strong> is:<\/p>\n\n\n\n [ Converting to percentages:<\/p>\n\n\n\n [ Since Mendel\u2019s expected percentage of green peas (75%) is within this confidence interval, we cannot conclude that the experimental results contradict Mendel\u2019s theory<\/strong>. The variation observed is likely due to natural random sampling variability rather than an error in Mendel’s predictions.<\/p>\n","protected":false},"excerpt":{"rendered":" One of Mendel\u2019s famous genetics experiments yielded 580 peas, with 428 of them green and 152 yellow. a. Find a 99% confidence interval estimate of the percentage of green peas. b. Based on his theory of genetics, Mendel expected that 75% of the offspring peas would be green. Given that the percentage of offspring green […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-187332","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187332","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=187332"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187332\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187332"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187332"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187332"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Answers:<\/h3>\n\n\n\n
(a) 99% Confidence Interval for the Percentage of Green Peas:<\/h4>\n\n\n\n
(b) Do the results contradict Mendel\u2019s theory?<\/h4>\n\n\n\n
\n\n\n\nExplanation:<\/h3>\n\n\n\n
\\hat{p} = \\frac{428}{580} \\approx 0.7379 \\text{ or } 73.79\\%
]<\/p>\n\n\n\n
SE = \\sqrt{\\frac{\\hat{p} (1 – \\hat{p})}{n}} = \\sqrt{\\frac{(0.7379)(1 – 0.7379)}{580}} \\approx 0.0188
]<\/p>\n\n\n\n
ME = Z \\times SE = 2.576 \\times 0.0188 \\approx 0.0487
]<\/p>\n\n\n\n
\\hat{p} \\pm ME = 0.7379 \\pm 0.0487 = (0.6909, 0.7849)
]<\/p>\n\n\n\n
(69.09\\%, 78.50\\%)
]<\/p>\n\n\n\n