MATH 302 University of Arizona Global Campus Statistics Questions

Normal Distributions are the most common distributions in statistics. If a randomvariable X is normally distributed with a mean μ and a standard deviation σ.
X ~ N(μ, σ) ; Z ~ N(0, 1)
Normal distributions are known as “bell-shaped curve”
To find the probabilities of normal distributions using a Normal Distribution Table,
we would start by converting the x values to a standard normal z-curve. The
equation of the z – score;
𝑥−𝜇
𝑧=
𝜎
Nowadays, we do not need to do this conversion to the standard normal
distribution, since Excel does it automatically for us.
Excel can only find Less Than probabilities, therefore it is important to make sure
that your problem is only including the less than inequality ( 25,000) = 1 – NORM.DIST(25000,25650,1560.09,TRUE)
In Excel make sure you hit the “=“ sign first, then the 1 – and then start typing in
NORM.DIST(. From here make sure you include the left parenthesis then type in the
x value, the mean, the standard deviation, then either True. Then close the
parenthesis ) and hit Enter.
The probability that the sample mean for the new sample of 5 cars is below
$25,000 is 66.15%.
Remember: Once you hit “Enter” the answer returns a decimal. You need to
convert it to a percentage if you want to read a percentage.
3. Assume that 5 additional cars are randomly sampled, and their prices are
recorded. What is the probability that the sample mean price of the 5 new cars will
be between $24,000 and $25,000?
Because of the word “between”, we want to find this probability
P(24000 < 𝑋̅< 25000). Since we are using the same data the mean and the new SD will be the same. Remember the function in Excel are in the less than form. This means we will need to do an extra step in Excel to get the probability we want. P(24000 < 𝑋̅ < 25000) = P(𝑋̅ < 25000) – P(𝑋̅ < 24000) = NORM.DIST(25000, 25650, 1560.09,TRUE) - NORM.DIST(24000,25650,1560.09,TRUE) In Excel make sure you hit the “=“ sign first, then start typing in NORM.DIST(. From here make sure you include the left parenthesis then type in the x value, the mean, the standard deviation, then either True. Then close the parenthesis ), hit the minus – sign then Repeat and then hit Enter. The probability that the sample mean for the new sample of 5 cars is between $24,000 and $25,000 is 19.34%. Remember: Once you hit “Enter” the answer returns a decimal. You need to convert it to a percentage if you want to read a percentage. Vehicle type/Class Year CAR 2017 CAR 2018 CAR 2019 CAR 2020 CAR 2015 CAR 2020 CAR 2019 CAR 2019 CAR 2020 CAR 2018 Qualitative Qualitative Make BMW Volkswagen Honda Subaru Mitsubishi Ford Chevrolet Nissan Dodge Audi Qualitative Model M2 Golf R Civic Type R WRX STI Lancer Evo Mustang GT Camaro 2SS 370Z Nismo Charger Scat TT Qualitative Mean: Median: SD: Sample size: Price $40.999 $40.000 $41.498 $40.435 $45.198 $44.095 $43.895 $42.990 $45.000 $46.247 Quantitative $43.036 $43.443 2190,91174 10 MPG (city) 21 21 22 16 17 15 16 17 15 23 Quantitative MPG(highway) 26 29 28 22 23 24 24 26 24 30 Quantitative Horsepower 365 hp 292 hp 306 hp 310 hp 303 hp 460 hp 455 hp 350 hp 485 hp 220 hp Quantitative Price Mean 43035,7 Standard Error692,827124 Median 43442,5 Mode #N/A Standard Deviation 2190,91174 Sample Variance 4800094,23 Kurtosis -1,512995 Skewness -0,0663174 Range 6247 Minimum 40000 Maximum 46247 Sum 430357 Count 10 Confidence Level(95.0%) 1567,28384