Question

Using the daily high and low temperature readings at Chicago's O'Hare International Airport for an entire year, a meteorologist made a scatterplot rel

Normal distributions
ANSWERED
asked 2020-12-28

Using the daily high and low temperature readings at Chicago's O'Hare International Airport for an entire year, a meteorologist made a scatterplot relating \(y = high\) temperature to \(x = low\) temperature, both in degrees Fahrenheit.
After verifying that the conditions for the regression model were met, the meteorologist calculated the equation of the population regression line to be \(\displaystyle{\left[\mu_{{y}}={16.6}+{1.02}\right]}{w}{i}{t}{h}{\left[\sigma={6.6}+^{\circ}{F}\right]}\).
About what percent of days with a low temperature of \(\displaystyle{40}^{\circ}\) F have a high temperature greater than \(\displaystyle{70}^{\circ}\) F?

Answers (1)

2020-12-29

Step 1
Given:
\(\displaystyle{\left[\mu_{{y}}={16.6}+{1.02}{x}{r}{i}{>}{h}\right]}\) (Equation population regression line)
\(\displaystyle{\left[\sigma={6.64}\right]}\)
The average high temperature on days where the low temperature is \(\displaystyle{40}^{\circ}\) F according to the population regression line can be found by replacing 2 in the regression line equation by 40 and evaluating.
\(\displaystyle{\left[\mu_{{y}}={16.6}+{1.02}{\left({40}\right)}={16.6}+{40.8}={57.4}\right]}\)
Thus the mean is 57.4 and the standard deviation is 6.64.
Since the conditions are met, the response y varies according to a Normal distribution.
The z-score is the value decreased by the mean, divided by the standard deviation.
\(\displaystyle{\left[{z}={\frac{{{x}-\mu}}{{\sigma}}}={\frac{{{70}-{57.4}}}{{{6.64}}}}\approx{1.90}\right]}\)
Determine the corresponding probability using the normal probability table in the appendix. \(\displaystyle{\left[{P}{\left({Z}{<}{1.90}\right)}\right]}\) is given in the row starting with 1.9 and in the column starting with .00 of the standard normal probability table in the appendix.
\(P(X>70)=P(Z>1.90)\)
\(=1-P(Z<1.90)\)
\(=1-0.9713\)
\(=0.0287\)
\(=2.87\%\)
Thus about 2.87% of the days with a low temperature of \(\displaystyle{40}^{\circ}\) F are expected to have a high temperature that is greater than \(\displaystyle{70}^{\circ}\) F.
Result: 2.87%

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-07-04

Using the daily high and low temperature readings at Chicago's O'Hare International Airport for an entire year, a meteorologist made a scatterplot relating y = high temperature to x = low temperature, both in degrees Fahrenheit.

After verifying that the conditions for the regression model were met, the meteorologist calculated the equation of the population regression line to be  \(\left[\mu_y=16.6+1.02\right] with \left[\sigma = 6.6+^\circ F\right]\)

If the meteorologist used a random sample of 10 days to calculate the regression line instead of using all the days in the year, would the slope of the sample regression line be exactly 1.02? Explain your answer.

asked 2021-01-10

Using the health records of ever student at a high school, the school nurse created a scatterplot relating y = height (in centimeters) to x = age (in years).
After verifying that the conditions for the regression model were met, the nurse calculated the equation of the population regression line to be \(\displaystyle\mu_{{0}}={105}+{4.2}{x}\ \text{with}\ \sigma={7}{c}{m}\).
About what percent of 15-year-old students at this school are taller than 180 cm?

asked 2021-06-04
Suppose that you want to perform a hypothesis test to compare several population means, using independent samples. In each case, decide whether you would use the one-way ANOVA test, the Kruskal-Wallis test, or neither of these tests. Preliminary data analyses of the samples suggest that the distributions of the variable are not normal and have quite different shapes.
asked 2021-06-07

1) If A and B are mutually exclusive events with \(P(A) = 0.3\) and \(P(B) = 0.5\), then \(P(A\cap B)=?\)
2) An experiment consists of four outcomes with \(P(E_1)=0.2,P(E_2)=0.3,\) and \(P(E_3)=0.4\). The probability of outcome \(E_4\) is ?
3) If A and B are mutually exclusive events with \(P(A) = 0.3\) and \(P(B) = 0.5\), then \(P(A\cap B) =?\)
4) The empirical rule states that, for data having a bell-shaped distribution, the percentage of data values being within one standard deviation of the mean is approximately
5) If a penny is tossed four times and comes up heads all four times, the probability of heads on the fifth trial is ?

...