Question

question 9 (1 point)
give an exact answer based on the empirical rule for the question given below.
the price paid for a particular model of hd television is approximately a normal distribution. the mean price paid is $1400 and the standard deviation is $135.
what is the approximate percentage of buyers who paid between $1400 and $1,535?
your answer:

answer
question 10 (1 point)
a machine fills 10lb bags of sugar. 99.7% of the bags have weights between 9.897 lbs and 10.407 lbs.
assuming this data for the weight of the bags follows a normal distribution find the mean and standard deviation of that distribution.
give exact results.
mean:

standard deviation:

Explanation:

Question 9

Step1: Calculate the number of standard deviations

The formula for the \(z -\)score is \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 1400\), \(\sigma=135\), and \(x = 1535\).

$$z=\frac{1535 - 1400}{135}=\frac{135}{135}=1$$

Step2: Apply the empirical rule

The empirical rule for a normal distribution states that about \(68\%\) of the data lies within \(z=\pm1\) (i.e., \(\mu\pm\sigma\)), about \(95\%\) lies within \(z = \pm2\) (i.e., \(\mu\pm2\sigma\)), and about \(99.7\%\) lies within \(z=\pm3\) (i.e., \(\mu\pm3\sigma\)). The area between \(z = 0\) (the mean) and \(z = 1\) is half of the area between \(z=- 1\) and \(z = 1\).
Since the area between \(z=-1\) and \(z = 1\) is \(68\%\), the area between \(z = 0\) and \(z=1\) is \(\frac{68\%}{2}=34\%\)

Step1: Use the property of the empirical rule for \(99.7\%\) of the data

For a normal distribution, \(99.7\%\) of the data lies within \(\mu\pm3\sigma\). If the lower bound is \(L = 9.897\) and the upper bound is \(U=10.407\), then the mean \(\mu=\frac{L + U}{2}\)

$$ \mu=\frac{9.897+10.407}{2}=\frac{20.304}{2}=10.152 $$

Step2: Calculate the standard deviation

We know that \(U=\mu + 3\sigma\) (or \(L=\mu-3\sigma\)). Using \(U=\mu + 3\sigma\), we can solve for \(\sigma\). Substitute \(\mu = 10.152\) into \(U=\mu + 3\sigma\)

$$ 10.407=10.152+3\sigma $$

$$ 3\sigma=10.407 - 10.152 $$

$$ 3\sigma=0.255 $$

$$ \sigma=\frac{0.255}{3}=0.085 $$

Answer:

\(34\%\)

Question 10