To construct a 90% confidence interval estimate for the population mean, we use the formula for a confidence interval:Confidence Interval=x\u02c9\u00b1Z\u00d7sn\\text{Confidence Interval} = \\bar{x} \\pm Z \\times \\frac{s}{\\sqrt{n}}Confidence Interval=x\u02c9\u00b1Z\u00d7n\u200bs\u200b<\/p>\n\n\n\n
Where:<\/p>\n\n\n\n
- \n
- x\u02c9\\bar{x}x\u02c9 = sample mean<\/li>\n\n\n\n
- ZZZ = Z-score corresponding to the desired confidence level (for 90%, the Z-value is 1.645)<\/li>\n\n\n\n
- sss = sample standard deviation<\/li>\n\n\n\n
- nnn = sample size<\/li>\n<\/ul>\n\n\n\n
Steps:<\/h3>\n\n\n\n
- \n
- Identify the sample mean (x\u02c9\\bar{x}x\u02c9)<\/strong>: This is the average of your sample data points.<\/li>\n\n\n\n
- Identify the sample standard deviation (sss)<\/strong>: This measures the spread of your data points around the sample mean.<\/li>\n\n\n\n
- Identify the sample size (nnn)<\/strong>: The number of data points in your sample.<\/li>\n\n\n\n
- Z-value for a 90% confidence interval<\/strong>: The Z-score for a 90% confidence level is 1.645 (this corresponds to 5% in each tail of the standard normal distribution).<\/li>\n\n\n\n
- Calculate the standard error<\/strong>: This is calculated as:<\/li>\n<\/ol>\n\n\n\n
SE=snSE = \\frac{s}{\\sqrt{n}}SE=n\u200bs\u200b<\/p>\n\n\n\n
- \n
- Calculate the margin of error<\/strong>: Multiply the standard error by the Z-value.<\/li>\n<\/ol>\n\n\n\n
Margin of Error=Z\u00d7SE\\text{Margin of Error} = Z \\times SEMargin of Error=Z\u00d7SE<\/p>\n\n\n\n
- \n
- Construct the confidence interval<\/strong>: Finally, the confidence interval is:<\/li>\n<\/ol>\n\n\n\n
x\u02c9\u00b1Margin of Error\\bar{x} \\pm \\text{Margin of Error}x\u02c9\u00b1Margin of Error<\/p>\n\n\n\n
Example:<\/h3>\n\n\n\n
Suppose you are given the following values:<\/p>\n\n\n\n
- \n
- Sample mean (x\u02c9\\bar{x}x\u02c9) = 50<\/li>\n\n\n\n
- Sample standard deviation (sss) = 10<\/li>\n\n\n\n
- Sample size (nnn) = 25<\/li>\n<\/ul>\n\n\n\n
Using the formula:SE=1025=105=2SE = \\frac{10}{\\sqrt{25}} = \\frac{10}{5} = 2SE=25\u200b10\u200b=510\u200b=2<\/p>\n\n\n\n
The margin of error:Margin of Error=1.645\u00d72=3.29\\text{Margin of Error} = 1.645 \\times 2 = 3.29Margin of Error=1.645\u00d72=3.29<\/p>\n\n\n\n
The confidence interval will be:50\u00b13.2950 \\pm 3.2950\u00b13.29<\/p>\n\n\n\n
So, the confidence interval is:[46.71,53.29][46.71, 53.29][46.71,53.29]<\/p>\n\n\n\n
Thus, we are 90% confident that the population mean lies between 46.71 and 53.29.<\/p>\n\n\n\n
This confidence interval provides a range of values within which we expect the true population mean to fall, with a 90% level of confidence.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-1032.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"Construct a 90% confidence interval estimate for the population mean given the following values: The Correct Answer and Explanation is: To construct a 90% confidence interval estimate for the population mean, we use the formula for a confidence interval:Confidence Interval=x\u02c9\u00b1Z\u00d7sn\\text{Confidence Interval} = \\bar{x} \\pm Z \\times \\frac{s}{\\sqrt{n}}Confidence Interval=x\u02c9\u00b1Z\u00d7n\u200bs\u200b Where: Steps: SE=snSE = \\frac{s}{\\sqrt{n}}SE=n\u200bs\u200b Margin of Error=Z\u00d7SE\\text{Margin of Error} = […]<\/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-247576","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/247576","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=247576"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/247576\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=247576"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=247576"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=247576"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Construct the confidence interval<\/strong>: Finally, the confidence interval is:<\/li>\n<\/ol>\n\n\n\n
- Identify the sample standard deviation (sss)<\/strong>: This measures the spread of your data points around the sample mean.<\/li>\n\n\n\n
- Identify the sample mean (x\u02c9\\bar{x}x\u02c9)<\/strong>: This is the average of your sample data points.<\/li>\n\n\n\n