the mean temperature in new york city over last month was 84 degrees fahrenheit with the standard deviation of 8. assume that the distribution of nyc temperature is unimodal and symmetric.
a) using the empirical rule, fill out the followings.
about 68% of the last month, the temperature in nyc was between
and
.
about 95% of the last month, the temperature in nyc was between
and
.
for almost all days in the last month, the temperature in nyc was between
and
.
b) if the temperature in nyc on a day was 80, would this be considered unusual? explain.
no, 80 would not be considered unusual because it falls within one standard deviation of the mean.
yes, 80 would be considered unusual because it falls within one standard deviation of the mean.
c) if the temperature in nyc on a day was 66, would this be considered unusual? explain.
yes, 66 would be considered unusual because it is more than two standard deviations below the mean.
no, 66 would not be considered unusual because it is not more than two standard deviations from the mean.

Question

Accepted Answer

# Explanation:
## Step1: Apply the Empirical Rule for 68%
The Empirical Rule states that about 68% of the data lies within $1$ standard deviation ($\sigma$) of the mean ($\mu$).
$$
\begin{align*}
\mu-\sigma&=84 - 8=76\
\mu+\sigma&=84 + 8=92
\end{align*}
$$
## Step2: Apply the Empirical Rule for 95%
About 95% of the data lies within $2$ standard deviations of the mean.
$$
\begin{align*}
\mu - 2\sigma&=84-2\times8=84 - 16 = 68\
\mu+2\sigma&=84 + 2\times8=84+16 = 100
\end{align*}
$$
## Step3: Apply the Empirical Rule for almost all (99.7%)
Almost all (99.7%) of the data lies within $3$ standard deviations of the mean.
$$
\begin{align*}
\mu-3\sigma&=84-3\times8=84 - 24=60\
\mu + 3\sigma&=84+3\times8=84 + 24=108
\end{align*}
$$
## Step4: Check if 80 is unusual
Calculate the difference from the mean: $|84 - 80|=4$. Since $4<8$ (one - standard deviation $=8$), 80 is within one standard deviation of the mean.
## Step5: Check if 66 is unusual
Calculate the difference from the mean: $|84 - 66| = 18$. Since $18>16$ ($2\sigma=16$), 66 is more than two standard deviations below the mean.

# Answer:
a) About 68%: $76$ and $92$; About 95%: $68$ and $100$; Almost all: $60$ and $108$
b) No, 80 would not be considered unusual because it falls within one standard deviation of the mean.
c) Yes, 66 would be considered unusual because it is more than two standard deviations below the mean.

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the Empirical Rule for 68%

Step2: Apply the Empirical Rule for 95%

Step3: Apply the Empirical Rule for almost all (99.7%)

Step4: Check if 80 is unusual

Step5: Check if 66 is unusual

Answer: