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The fill amount of bottles of a soft drink is normally distributed with a mean of 2.0 liters and a standard deviation of 0.05 liters. If you select a random sample of 25 bottles, what is the probability that the sample mean will be below 1.98 liters?NN vs nn In statistics, two of the symbols that are commonly used are <span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="N" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">NN and <span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="n." role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">n.n. While others may simply misunderstand these as merely part of the English alphabet, these symbols are used to represent the number of population and sample. <span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="N" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">NN is for the number of population, and <span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="n" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">nn is the number of sample.
Answer and Explanation:In this problem, the given are: - Population mean, μ=2.0μ=2.0
- Population standard deviation, σ=0.05σ=0.05
Now, to solve the problem, we need to look for the z-score of <span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="X¯=1.98" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">¯X=1.98X¯=1.98 using the formula <span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="z=X¯−μσn." role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">z=¯X−μσ√n.z=X¯−μσn. Substituting the necessary values, we get <span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="z=X¯−μσn=1.98−2.00.0525z=−2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">z=¯X−μσ√n=1.98−2.00.05√25z=−2z=X¯−μσn=1.98−2.00.0525z=−2 Now, using a z-table, we find that <span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="P(z<−2)=0.02275." role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.94px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">P(z<−2)=0.02275.P(z<−2)=0.02275. Therefore, the probability that the sample mean will be below 1.98 liters is 0.02275 or 2.275%.
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