[APPLET] The annual earnings (in dollars) for 30 randomly sele...

Question

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)

48,69446,85642,91261,67271,11254,861

69,45471,84159,75169,61254,28452,166

66,36048,16465,27235,25061,12765,397

58,92558,91659,01753,07045,19969,941

69,49257,08553,82952,69268,29853,792

Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The annual salaries in units of dollars for 10 randomly chosen dental hygienists are listed below. Assume the population is normally distributed. 52,340, 54,120, 47,890, 60,210, 63,500, 59,780, 57,430, 61,200, 55,670. And 58,900 construct a 95% confidence interval for the population mean annual salary or dental hygienist round your answer to the nearest whole number. Awesome. So it appears for this particular prong we're asked. To create a 95% confidence interval for the population means annual salary for this particular group of randomly chosen dental hygienists, and then we're asked to round our answer to the nearest whole number. So with that in mind, now that we know that we're trying to create this 95% confidence interval, our first step is we need to find the sample mean, which is X bar. So, let us recall and note that X bar is equal to the sum of X subscript I divided by n where n in this case the sample size is 10 means we're dealing with 10 randomly chosen dental hygienists. So now our next step is we need to find the sample standard deviation S, which, as we should recall, S is equal to the square root of the sum of parentheses X subscript I minus X to the power 2 divided by N minus 1. Awesome. So with that in mind, in order to help us solve for our answer, let's create a table to organize all of our data. And when we create a table, it should look something like the following. So we have our different values for I, so we have 123456789, and 10 for our 10 randomly chosen dental hygienists. Then we have our column for the values of X subscript I. Then we have our column for X subscript I minus X bar. Then we have our value for X subscript i minus X bar to the power of 2. And we should note that N minus 1 will be equal to 9, means we have 10 minus 1 will be equal to 9. And we should note that the sum of XI will be equal to 571,040 and X bar will be equal to 57,104. And we should note that the sum of X subscript I minus X to the power 2 will be. 196,380,240. And S will be equal to. 4,671.19114. And once again I've gone ahead and plugged all of our data points into an Excel spreadsheet and let the spreadsheet do all the math for you. And when you do, you will receive the following results. So with that in mind, moving right along here, from the table, we will determine once again that X bar. is equal to 57,104 and S will be equal to 4,671.19114. And since the sample size is small, meaning that N equals 10 is less than 30, and the population standard deviation is unknown to us. Because of this fact, we can recall and use the T distribution. So with that in mind, we need to find the criticalt value for a 95% confidence, and we also need to determine the degrees of freedom. So with that in mind, let us recall that the degrees of freedom, which we'll call DF for short, DF is equal to N minus 1, which is equal to 10 minus 1, which is equal to 9. And our critical T value, which we'll call T to the power of star, and T to the power of star. is approximately equal to 2.262. Fantastic. And we can find our critical T value by using a table from your textbook. Awesome. So moving right along here, our next step now is we need to calculate the standard error, which we'll call SE for short, and SE, the standard error, is equal to S divided by the square root of N, which is equal to 4,671.19114 divided by the square root of 10. Which when we plug it into our calculator, it will be approximately equal to 1,477.160339. So now, moving right along here, our next step is we need to calculate the margin of error, which will denote as E. So E, the margin of error is equal to T to the power of star. Multiplied by SE, which is equal to when we plug in our known variables, 2.262 multiplied by 1,477.160339, which will mean that our margin of error E is approximately equal to 3,341. Fantastic. So once again, 3,341. So now, our next step is we need to construct our confidence interval, which, as we should recall, states that X is, so X plus or minus E is equal to 57,104 plus or minus 3,341, which will mean that we could go ahead and write parentheses. 57,104 minus 3,341. Comma Which is in parentheses, so parentheses 57,104 minus 3,341 57,104 plus 3,341. Fantastic. And don't forget your other set of parentheses. So that means when we perform that calculation, we will find that it's that X bar plus or minus E. Will be equal to parentheses 53,763, 60,445. So that will mean that our final answer without a doubt will have to be that the 95% confidence interval for the population mean annual salary for dental hygienists is, and don't forget your dollar symbols, visits and units of dollar. will be $53,763 60,445 dollars. So to go back to look at our prom itself to quickly recap what we've learned, So it turns out once again. That the 95% confidence interval for the population mean annual salary for dental hygienists, without a doubt, will have to be, once again, Parentheses. $53,763 60,445 dollars. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Normal Distribution

Sample Mean and Standard Deviation

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