Math  /  Data & Statistics

QuestionDance Company Students The number of students who belong to the dance company at each of several randomly selected small universities is shown below. Round sample statistics and final answers to at least one decimal place. \begin{tabular}{llllllll} 27 & 21 & 47 & 21 & 32 & 32 & 35 & 29 \\ 35 & 26 & 35 & 30 & 21 & 21 & 32 & 25 \end{tabular}
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Estimate the true population mean size of a university dance company with 95%95 \% confidence. Assume the variable is normally distributed.

Studdy Solution

STEP 1

1. The data set contains the number of students in dance companies at several small universities.
2. The sample is normally distributed.
3. We are estimating the population mean with a 95% confidence interval.
4. The sample size is n=16 n = 16 .

STEP 2

1. Calculate the sample mean and sample standard deviation.
2. Determine the standard error of the mean.
3. Find the critical value for a 95% confidence interval.
4. Calculate the confidence interval for the population mean.

STEP 3

Calculate the sample mean (xˉ\bar{x}):
List the data points:
27,21,47,21,32,32,35,29,35,26,35,30,21,21,32,25 27, 21, 47, 21, 32, 32, 35, 29, 35, 26, 35, 30, 21, 21, 32, 25
Calculate the sum of the data points:
Sum=27+21+47+21+32+32+35+29+35+26+35+30+21+21+32+25=424 \text{Sum} = 27 + 21 + 47 + 21 + 32 + 32 + 35 + 29 + 35 + 26 + 35 + 30 + 21 + 21 + 32 + 25 = 424
Calculate the sample mean:
xˉ=Sumn=42416=26.5 \bar{x} = \frac{\text{Sum}}{n} = \frac{424}{16} = 26.5

STEP 4

Calculate the sample standard deviation (ss):
First, find the deviations from the mean, square them, and sum them:
(xixˉ)2=(2726.5)2+(2126.5)2++(2526.5)2 \sum (x_i - \bar{x})^2 = (27-26.5)^2 + (21-26.5)^2 + \ldots + (25-26.5)^2
=0.25+30.25+420.25+30.25+42.25+42.25+72.25+6.25+72.25+0.25+72.25+12.25+30.25+30.25+42.25+2.25=420.5 = 0.25 + 30.25 + 420.25 + 30.25 + 42.25 + 42.25 + 72.25 + 6.25 + 72.25 + 0.25 + 72.25 + 12.25 + 30.25 + 30.25 + 42.25 + 2.25 = 420.5
Calculate the sample standard deviation:
s=(xixˉ)2n1=420.5155.3 s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{420.5}{15}} \approx 5.3

STEP 5

Determine the standard error of the mean (SE):
SE=sn=5.316=5.34=1.325 SE = \frac{s}{\sqrt{n}} = \frac{5.3}{\sqrt{16}} = \frac{5.3}{4} = 1.325

STEP 6

Find the critical value for a 95% confidence interval:
For a 95% confidence level and n1=15 n-1 = 15 degrees of freedom, the critical value (t t^* ) from the t-distribution table is approximately 2.131.

STEP 7

Calculate the confidence interval for the population mean:
The margin of error (ME) is:
ME=t×SE=2.131×1.3252.82 ME = t^* \times SE = 2.131 \times 1.325 \approx 2.82
The confidence interval is:
xˉ±ME=26.5±2.82 \bar{x} \pm ME = 26.5 \pm 2.82
=(23.68,29.32) = (23.68, 29.32)
The 95% confidence interval for the true population mean size of a university dance company is:
(23.7,29.3) \boxed{(23.7, 29.3)}

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