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Solve the problem_Human body temperatures are normally distributed with mean of 98.20F and standard deviation of 0.62*F. If 30 people are randomly selected, find th...

Question

Solve the problem_Human body temperatures are normally distributed with mean of 98.20F and standard deviation of 0.62*F. If 30 people are randomly selected, find the probability that their mean body temperature will be more than 98.35FF

Solve the problem_ Human body temperatures are normally distributed with mean of 98.20F and standard deviation of 0.62*F. If 30 people are randomly selected, find the probability that their mean body temperature will be more than 98.35FF



Answers

Consider a normal distribution with mean 30 and standard deviation $2 .$ What is the probability a value selected at random from this distribution is greater than $30 ?$

So we're gonna be applying Chevy Chef serum to the temperatures. Uh since it's a bell shaped distribution were allowed to do this um And let's get started. So the mean is going to be 98.2. Yeah, and that's going to be in degrees Fahrenheit. Some people put it right up there And our sigma are standard deviation is going to be 0.62. Our Chevy Chevy five Theorem application will be to find out how much of our data is going to be within three standard deviations of the mean. So that's gonna be one minus one of our case square where K is the number of standard deviations we have. So that's one And this 1/3 squared. So we're gonna have 1 -1/9 or 8/9 Which is equal to a percentage of 89%. And then we can find out the upper and lower bounds simply by taking our average of 98.2 mhm Adding and subtracting the three standard deviations of 062 mm. Yeah. To get our upper and lower bounds, so for our lower This is going to be equal to 96.34 degrees Fahrenheit. And for upper bounds This is going to be 106 degrees Fahrenheit

Hands. Claris. When you right here. So we're giving her mean last 98.25 Fahrenheit and standard aviation this point fought 75 burning. So, the party, we're gonna find the 90% tall. So we have access people, too. I mean, plus C standard aviation. We spend a normal distribution table, but we work backwards. So for 99 0.21 does this 90th percentile source The volume is 1.22 for our fifth percentile. We know that 5% it's gonna give us a value of C is equal to negative 1.6 pork. Why? So you get 97.2 and reports seen it separates the coolest. 25% says he calls negative 250.67 And since 0.2514 value is the table to the closest 2.25 So blink to 51 fork. We're gonna use our C formula see formula, which is a born too negative 0.67 Just legal to X minus. I needed 0.25 over 175 Get X is equal to 97.75

Program 44. Given the mean understand their division off body temperatures off healthy adults ex bore the mean, which equals 98.2 39 degrees. And the standard division is equals 4.6, 2000 and nine degrees. And this follows a bill shaped distribution. The data collected follows a bell shaped distribution like this. We want to use chips. Ethereum toe. Say something about the percentage of healthy adults with body temperatures between two standard deviations off the meat. Then we can use chips with er, um, to get the minimum percentage off adults. That law is between two standard divisions or case under division. If we have here X bar in the middle, then if we have here X bar plus key and we have here X bar minus key, he s okay. We have your KS, which means the division from the mean by a value office, we can calculate meet the minimum percentage. That lawyer is in this range and the memory percentage given by ship shift serum equals one minus one. Divided by case square multiplied by 100 as a percentage, then it equals one minus one, divided by key is to Then we have one divided by four multiplied by 100 which gives 75%. And to calculate the minimum and the maximum value, we will calculate this value and its value. The lower name, it or the minimum. We can say that the minimum can see it a minimum. The minimum value within two Standard division equals X bar minus two s, which equals 98.2 minus two multiplied by s. And this equals 96 0.96 very 90 degrees and the maximum value within two standard divisions X bar plus two multiplied by s equals 98.2 minus two are applied by s sorry. Plus two multiplied by s equals 99.44 59 degrees. And this means that the minimum percentage that has body temperatures between this value and this value equals 75%. And this is the final answer off our problem

Behold this glorious, normal bell shaped curve right there in the middle is going to be our average new one. Standard deviation now from it is going to be in red, which is going to have 68% of all of our data, We go on another standard deviation two. Standard deviations away from the mean and that's going to have 95% of our data. This is where the 5% of unusualness comes from. If something is 5% chance of not happening or happening is considered to be unusual. And outside of those two standard deviations, As for the third standard deviation, that's going to have 99.5 of all of our data. So, our first question asked, Yeah, What is the approximate percentage of healthy adults with body temperatures that are within one standard deviation of the mean? Or between 97.58 and 98.82, why don't we put those bounds there? And Red 97 58 And 98.82. Yeah. So we are also given that the mean is 98.2°F. Yeah. And a standard deviation of 062. Mhm Yeah, Yeah. So now we can get to finally answering your question and for one standard deviation this is going to allow for 68% of our data. Yeah. And then for part B between 96.34 and 100.06°F. Now this seems a little high for the 100.6, considering that we can only go up by increments of .62. So I'm gonna guess that that's going to be three standard deviations away. So what we can do to verify this is take her average which is 98.2 and add that to three times are standard deviation, So that's gonna be three times 0.62 mm. And what we end up getting is exactly 100.06. Mhm. Mhm. And I made a small mistake on copying are 99.5 was supposed to be 99.7, so this is going to encompass 99.7% of the graph when we go from 96.34°F and 100.06.


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