Solved on Dec 17, 2023

Find the standard error of the sample mean for the daily high temperatures (in $\degree$ F) in Des Moines over 1 week: Monday (64.5), Tuesday (64), Wednesday (66.5), Thursday (64), Friday (62.5), Saturday (61), Sunday (63).

STEP 1

Assumptions
1. The temperatures given are the daily high temperatures for Des Moines for a week.
2. We are to calculate the standard error of the sample mean.
3. The standard error of the sample mean is given by the formula: $SE = \frac{s}{\sqrt{n}}$ where $s$ is the sample standard deviation and $n$ is the sample size.

STEP 2

First, we need to calculate the sample mean temperature. The sample mean is given by the formula: $\bar{x} = \frac{\sum{x_i}}{n}$ where $x_i$ represents each temperature value and $n$ is the number of temperature values.

STEP 3

Now, plug in the given temperature values to calculate the sample mean.
$\bar{x} = \frac{64.5 + 64 + 66.5 + 64 + 62.5 + 61 + 63}{7}$

STEP 4

Calculate the sample mean temperature.
$\bar{x} = \frac{445.5}{7}$
$\bar{x} = 63.64$ (rounded to two decimal places)

STEP 5

Next, we need to calculate the sample variance. The sample variance $s^2$ is given by the formula: $s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n - 1}$

STEP 6

Calculate the squared differences between each temperature value and the sample mean.
$(64.5 - 63.64)^2 = 0.86^2 = 0.7396$ $(64 - 63.64)^2 = 0.36^2 = 0.1296$ $(66.5 - 63.64)^2 = 2.86^2 = 8.1796$ $(64 - 63.64)^2 = 0.36^2 = 0.1296$ $(62.5 - 63.64)^2 = -1.14^2 = 1.2996$ $(61 - 63.64)^2 = -2.64^2 = 6.9696$ $(63 - 63.64)^2 = -0.64^2 = 0.4096$

STEP 7

Sum the squared differences to find the numerator for the sample variance.
$\sum{(x_i - \bar{x})^2} = 0.7396 + 0.1296 + 8.1796 + 0.1296 + 1.2996 + 6.9696 + 0.4096$
$\sum{(x_i - \bar{x})^2} = 17.8572$

STEP 8

Now, divide the sum of squared differences by $n - 1$ to find the sample variance.
$s^2 = \frac{17.8572}{7 - 1}$
$s^2 = \frac{17.8572}{6}$
$s^2 = 2.9762$ (rounded to four decimal places)

STEP 9

Take the square root of the sample variance to find the sample standard deviation.
$s = \sqrt{s^2}$
$s = \sqrt{2.9762}$
$s = 1.7252$ (rounded to four decimal places)

STEP 10

Finally, calculate the standard error of the sample mean using the sample standard deviation and the sample size.
$SE = \frac{s}{\sqrt{n}}$
$SE = \frac{1.7252}{\sqrt{7}}$
$SE = \frac{1.7252}{2.6458}$ (rounded to four decimal places for the square root of 7)

STEP 11

Calculate the standard error of the sample mean.
$SE = 0.6519$ (rounded to four decimal places)
Rounded to the hundredths place, the standard error of the sample mean is approximately $0.65$ .

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Find the standard error of the sample mean for the daily high temperatures (in $\degree$ F) in Des Moines over 1 week: Monday (64.5), Tuesday (64), Wednesday (66.5), Thursday (64), Friday (62.5), Saturday (61), Sunday (63).

STEP 1

STEP 2

First, we need to calculate the sample mean temperature. The sample mean is given by the formula: $\bar{x} = \frac{\sum{x_i}}{n}$ where $x_i$ represents each temperature value and $n$ is the number of temperature values.

STEP 3

Now, plug in the given temperature values to calculate the sample mean.
$\bar{x} = \frac{64.5 + 64 + 66.5 + 64 + 62.5 + 61 + 63}{7}$

STEP 4

Calculate the sample mean temperature.
$\bar{x} = \frac{445.5}{7}$
$\bar{x} = 63.64$ (rounded to two decimal places)

STEP 5

Next, we need to calculate the sample variance. The sample variance $s^2$ is given by the formula: $s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n - 1}$

STEP 6

STEP 7

Sum the squared differences to find the numerator for the sample variance.
$\sum{(x_i - \bar{x})^2} = 0.7396 + 0.1296 + 8.1796 + 0.1296 + 1.2996 + 6.9696 + 0.4096$
$\sum{(x_i - \bar{x})^2} = 17.8572$

STEP 8

Now, divide the sum of squared differences by $n - 1$ to find the sample variance.
$s^2 = \frac{17.8572}{7 - 1}$
$s^2 = \frac{17.8572}{6}$
$s^2 = 2.9762$ (rounded to four decimal places)

STEP 9

Take the square root of the sample variance to find the sample standard deviation.
$s = \sqrt{s^2}$
$s = \sqrt{2.9762}$
$s = 1.7252$ (rounded to four decimal places)

STEP 10

Finally, calculate the standard error of the sample mean using the sample standard deviation and the sample size.
$SE = \frac{s}{\sqrt{n}}$
$SE = \frac{1.7252}{\sqrt{7}}$
$SE = \frac{1.7252}{2.6458}$ (rounded to four decimal places for the square root of 7)

STEP 11

Calculate the standard error of the sample mean.
$SE = 0.6519$ (rounded to four decimal places)
Rounded to the hundredths place, the standard error of the sample mean is approximately $0.65$ .

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Find the standard error of the sample mean for the daily high temperatures (in °\degree°F) in Des Moines over 1 week: Monday (64.5), Tuesday (64), Wednesday (66.5), Thursday (64), Friday (62.5), Saturday (61), Sunday (63).

STEP 1

STEP 2

First, we need to calculate the sample mean temperature. The sample mean is given by the formula: xˉ=∑xin \bar{x} = \frac{\sum{x_i}}{n} xˉ=n∑xi​​ where xi x_i xi​ represents each temperature value and n n n is the number of temperature values.

STEP 3

Now, plug in the given temperature values to calculate the sample mean.xˉ=64.5+64+66.5+64+62.5+61+637 \bar{x} = \frac{64.5 + 64 + 66.5 + 64 + 62.5 + 61 + 63}{7} xˉ=764.5+64+66.5+64+62.5+61+63​

STEP 4

Calculate the sample mean temperature.xˉ=445.57 \bar{x} = \frac{445.5}{7} xˉ=7445.5​xˉ=63.64 \bar{x} = 63.64 xˉ=63.64 (rounded to two decimal places)

STEP 5

Next, we need to calculate the sample variance. The sample variance s2 s^2 s2 is given by the formula: s2=∑(xi−xˉ)2n−1 s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n - 1} s2=n−1∑(xi​−xˉ)2​

STEP 6

STEP 7

STEP 8

Now, divide the sum of squared differences by n−1 n - 1 n−1 to find the sample variance.s2=17.85727−1 s^2 = \frac{17.8572}{7 - 1} s2=7−117.8572​s2=17.85726 s^2 = \frac{17.8572}{6} s2=617.8572​s2=2.9762 s^2 = 2.9762 s2=2.9762 (rounded to four decimal places)

STEP 9

Take the square root of the sample variance to find the sample standard deviation.s=s2 s = \sqrt{s^2} s=s2​s=2.9762 s = \sqrt{2.9762} s=2.9762​s=1.7252 s = 1.7252 s=1.7252 (rounded to four decimal places)

STEP 10

STEP 11

Calculate the standard error of the sample mean.SE=0.6519 SE = 0.6519 SE=0.6519 (rounded to four decimal places)Rounded to the hundredths place, the standard error of the sample mean is approximately 0.65 0.65 0.65.

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Find the standard error of the sample mean for the daily high temperatures (in $\degree$ F) in Des Moines over 1 week: Monday (64.5), Tuesday (64), Wednesday (66.5), Thursday (64), Friday (62.5), Saturday (61), Sunday (63).

First, we need to calculate the sample mean temperature. The sample mean is given by the formula: $\bar{x} = \frac{\sum{x_i}}{n}$ where $x_i$ represents each temperature value and $n$ is the number of temperature values.

Now, plug in the given temperature values to calculate the sample mean.
$\bar{x} = \frac{64.5 + 64 + 66.5 + 64 + 62.5 + 61 + 63}{7}$

Calculate the sample mean temperature.
$\bar{x} = \frac{445.5}{7}$
$\bar{x} = 63.64$ (rounded to two decimal places)

Next, we need to calculate the sample variance. The sample variance $s^2$ is given by the formula: $s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n - 1}$

Now, divide the sum of squared differences by $n - 1$ to find the sample variance.
$s^2 = \frac{17.8572}{7 - 1}$
$s^2 = \frac{17.8572}{6}$
$s^2 = 2.9762$ (rounded to four decimal places)

Take the square root of the sample variance to find the sample standard deviation.
$s = \sqrt{s^2}$
$s = \sqrt{2.9762}$
$s = 1.7252$ (rounded to four decimal places)

Calculate the standard error of the sample mean.
$SE = 0.6519$ (rounded to four decimal places)
Rounded to the hundredths place, the standard error of the sample mean is approximately $0.65$ .