Math / Data & Statistics

QuestionYou must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures: \begin{tabular}{|r|} \hline 17.5 \\ \hline 12.7 \\ \hline 12.7 \\ \hline 2.1 \\ \hline 7.4 \\ \hline 13.1 \\ \hline 39.6 \\ \hline 21.4 \\ \hline 33.9 \\ \hline 3.9 \\ \hline 25.1 \\ \hline 36.7 \\ \hline \end{tabular}
Find the $99 \%$ confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).
99\% C.I. =
Answer should be obtained without any preliminary rounding.

Studdy Solution

STEP 1

What is this asking? We need to find the range where we're 99% sure the true average temperature lies, based on a handful of temperature readings. Watch out! Don't mix up standard deviation with standard error – standard error is what we need for confidence intervals!
Also, remember to use the t-distribution since our sample size is small and we don't know the population's standard deviation.

STEP 2

1. Calculate the sample mean
2. Calculate the sample standard deviation
3. Calculate the standard error
4. Find the critical t-value
5. Calculate the margin of error
6. Calculate the confidence interval

STEP 3

Let's add up all the temperatures and divide by the number of readings to get the sample mean, often represented by $\bar{x}$ .
This gives us a good estimate of the average temperature.
$\bar{x} = \frac{17.5 + 12.7 + 12.7 + 2.1 + 7.4 + 13.1 + 39.6 + 21.4 + 33.9 + 3.9 + 25.1 + 36.7}{12}$ $\bar{x} = \frac{226.1}{12}$ $\bar{x} \approx 18.84$

STEP 4

Now, let's see how spread out our temperature readings are.
We'll calculate the sample standard deviation, denoted by $s$ , which tells us how much the individual temperatures typically differ from the mean.
$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$ Here, each $x_i$ is a temperature reading, $\bar{x}$ is our sample mean (approximately 18.84), and $n$ is the number of readings (12).
$s = \sqrt{\frac{(17.5 - 18.84)^2 + (12.7 - 18.84)^2 + ... + (36.7 - 18.84)^2}{12-1}}$ $s = \sqrt{\frac{1.7956 + 37.4544 + ... + 318.9796}{11}}$ $s = \sqrt{\frac{2098.8092}{11}}$ $s \approx \sqrt{190.80} \approx 13.81$

STEP 5

The standard error, denoted as $SE$ , measures how much our sample mean might differ from the true population mean.
It's calculated by dividing the sample standard deviation by the square root of the sample size.
$SE = \frac{s}{\sqrt{n}} = \frac{13.81}{\sqrt{12}} \approx \frac{13.81}{3.46} \approx 3.99$

STEP 6

Since we're dealing with a small sample and an unknown population standard deviation, we'll use the t-distribution.
For a 99% confidence level and 11 degrees of freedom ( $n - 1 = 12 - 1 = 11$ ), our **critical t-value**, which we can find in a t-table or using a calculator, is approximately 3.11.

STEP 7

The margin of error, denoted as $ME$ , tells us how much "wiggle room" we have around our sample mean.
We multiply the standard error by the critical t-value.
$ME = t \cdot SE = 3.11 \cdot 3.99 \approx 12.41$

STEP 8

Finally, we can calculate the confidence interval by taking our sample mean and adding/subtracting the margin of error.
$CI = (\bar{x} - ME, \bar{x} + ME)$ $CI = (18.84 - 12.41, 18.84 + 12.41)$ $CI = (6.43, 31.25)$

STEP 9

The 99% confidence interval is (6.43, 31.25).

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Studdy Solution

STEP 1

STEP 2

1. Calculate the sample mean
2. Calculate the sample standard deviation
3. Calculate the standard error
4. Find the critical t-value
5. Calculate the margin of error
6. Calculate the confidence interval

STEP 3

STEP 4

STEP 5

STEP 6

Since we're dealing with a small sample and an unknown population standard deviation, we'll use the t-distribution.
For a 99% confidence level and 11 degrees of freedom ( $n - 1 = 12 - 1 = 11$ ), our **critical t-value**, which we can find in a t-table or using a calculator, is approximately 3.11.

STEP 7

The margin of error, denoted as $ME$ , tells us how much "wiggle room" we have around our sample mean.
We multiply the standard error by the critical t-value.
$ME = t \cdot SE = 3.11 \cdot 3.99 \approx 12.41$

STEP 8

Finally, we can calculate the confidence interval by taking our sample mean and adding/subtracting the margin of error.
$CI = (\bar{x} - ME, \bar{x} + ME)$ $CI = (18.84 - 12.41, 18.84 + 12.41)$ $CI = (6.43, 31.25)$

STEP 9

The 99% confidence interval is (6.43, 31.25).

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Studdy Solution

STEP 1

STEP 2

1. Calculate the sample mean2. Calculate the sample standard deviation3. Calculate the standard error4. Find the critical *t*-value5. Calculate the margin of error6. Calculate the confidence interval

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

The **margin of error**, denoted as MEMEME, tells us how much "wiggle room" we have around our sample mean.We **multiply** the standard error by the critical *t*-value.ME=t⋅SE=3.11⋅3.99≈12.41ME = t \cdot SE = 3.11 \cdot 3.99 \approx 12.41ME=t⋅SE=3.11⋅3.99≈12.41

STEP 8

STEP 9

The 99% confidence interval is (6.43, 31.25).

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1. Calculate the sample mean
2. Calculate the sample standard deviation
3. Calculate the standard error
4. Find the critical t-value
5. Calculate the margin of error
6. Calculate the confidence interval

The margin of error, denoted as $ME$ , tells us how much "wiggle room" we have around our sample mean.
We multiply the standard error by the critical t-value.
$ME = t \cdot SE = 3.11 \cdot 3.99 \approx 12.41$