MATH302 University of Phoenix Week 2 Descriptive Statistics Discussion Recall the car data set you identified in Week 2. We know that this data set is norm

MATH302 University of Phoenix Week 2 Descriptive Statistics Discussion Recall the car data set you identified in Week 2. We know that this data set is normally distributed using the mean and SD you calculated. (Be sure you use the numbers without the supercar outlier)

For the next 4 cars that are sampled, what is the probability that the price will be less than $500 dollars below the mean? Make sure you interpret your results.

Please note: we are given a new sample size, we will need to calculate a new SD. Then, to find the value that is $500 below the mean you will need to take the mean and subtract $500 from it. For example, if the mean is $15,000 then $500 below this would be $14,500. Thus the probability you would want to find is P(x 16,000).

For the next 4 cars that are sampled, what is the probability that the price will be equal to the mean? Make sure you interpret your results. Use the same logic as above.

For the next 4 cars that are sampled, what is the probability that the price will be $1500 within the mean? Make sure you interpret your results. Use the same logic as above.

I encourage you to review the Week 4 normal probabilities PDF at the bottom of the forum. This will give you a step by step example to follow and show you how to find probabilities using Excel. I also encourage you to review the Week 4 Empirical Rule PDF. This will give you a better understanding on how to utilize the empirical rule.

Instructions: Your initial post should be at least 150 words. You must also respond to at least 2 other students. Responses should be a minimum of 50 words and may include direct questions. In your peer posts, compare the probabilities that you found with those of your classmates. Were they higher/lower and why? In your responses, refer to the specific data from your classmates’ posts. This week we will discuss the Empirical Rule.
The empirical rule allows you to determine the proximity of the data to the mean.
This only works for bell shape or symmetric distributions.
• The interval that is one standard deviation away contains approximately
68% of the data.
(??? ± 1 *SD)
• The interval that is two standard deviations away contains approximately
95% of the data.
(??? ± 2 *SD)
• The interval that is three standard deviations away contains approximately
99.7% of the data.
(??? ± 3 *SD)
Let’s continue to look at the Data from Week 2.
Car Price:
Observation 1
$
20,000
Observation 2
$
25,000
Observation 3
$
30,000
Observation 4
$
31,000
Observation 5
$
22,500
Observation 6
$
25,000
Observation 7
$
29,500
Observation 8
$
24,000
Observation 9
$
24,500
Observation 10 $
25,000
Mean:
$
25,650
Median:
$
25,000
SD:
$
3,488.47
Sample Size:
10
Using the Empirical Rule calculate how many data points fall within the 1, 2 and 3
SD’s?
1) First, we will need to calculate each interval.
25, 650 – 3,488.47 = $22,162 -> round to the nearest dollar
25, 650 + 3,488.47 = $29,138 -> round to the nearest dollar
The interval for approximal 68% of the data is ($22,162, $29,138).
But how many data points fall within this interval?
We see that observations, 1, 2, 5, 6, 8 ,9, and 10. 7 of the 10 observations fall
within this interval. That is
7
10
= 70% of the data falls within 1 SD.
This is very
close to 68%.
2) We will calculate the next interval.
25, 650 – (2) 3,488.47 = $18,673 -> round to the nearest dollar
25, 650 + (2) 3,488.47 = $32,627 -> round to the nearest dollar
The interval for approximal 95% of the data is ($18,673, $32,627).
But how many data points fall within this interval?
We see that observations, 1, 2, 3, 4 5, 6, 7 8 ,9, and 10. All 10 of the 10
observations fall within this interval. That is
10
10
= 100% of the data falls within 2
SD’s. This is very close to 95%. Since this is a smaller data set see that all the data
points fall within the first 2 observations is not uncommon. We would expect
results like this.
3) But it is still a good idea to calculate the last interval.
25, 650 – (3) 3,488.47 = $15,185 -> round to the nearest dollar
25, 650 + (3) 3,488.47 = $36,115 -> round to the nearest dollar
The interval for approximal 99.7% of the data is ($15,185, $36,115).
But how many data points fall within this interval?
Just like with the last interval all the data points fall within this interval and
because this is a small data set the results are as expected.
There are no data points that fall outside this range. There doesn’t appear to be
any outliers in this data set. We also see that the mean and median are close
together. There isn’t a big difference between the two values. Because of these
explanations, this data appears to be normal and have a normal distribution. The
data set does not seem to be skewed, in either direction.
We can see how the SD’s line up along the x-axis and it creates the bell-shaped
curve.
Descriptive statistics Discussion
For the initial data collected on the prices of 10 cars, the cars had an average price of
$126,276 and a standard deviation of $25,257.09. The 10 cars also had a median price of
$118,370. On the addition of an 11th car, a super car, the McLaren P1 worth $1879888 there was
a drastic change in some of the descriptive statistics. For the 11 cars the mean price was
$285,695 with a standard deviation of $529,276.62, while the median was $121,115.
Based on the statistics above we can observe that the addition of the supercar had a great
effect on the mean and standard deviation but had minimal effect on the median. The mean price
more than doubled by the addition of the supercar to the data while the standard deviation
increased by more than twenty times. The increase in standard deviation indicated that there was
greater variation in the modified dataset as compared to the original dataset. From the analysis,
the median was least affected by the addition of the supercar to the data. The median is
,therefore, best in representing the data as it is more central as compared to the mean and is least
affected by outliers. In the original dataset, the standard deviation was lower than the mean while
in the modified dataset the standard deviation was greater than the mean. The standard deviation
can, therefore, be used to detect outliers by comparing the value of standard deviation with the
mean. If standard deviation is greater than the mean then there is a high likelihood that the data
has outliers.
Descriptive statistics Discussion
For the initial data collected on the prices of 10 cars, the cars had an average price of
$126,276 and a standard deviation of $25,257.09. The 10 cars also had a median price of
$118,370. On the addition of an 11th car, a super car, the McLaren P1 worth $1879888 there was
a drastic change in some of the descriptive statistics. For the 11 cars the mean price was
$285,695 with a standard deviation of $529,276.62, while the median was $121,115.
Based on the statistics above we can observe that the addition of the supercar had a great
effect on the mean and standard deviation but had minimal effect on the median. The mean price
more than doubled by the addition of the supercar to the data while the standard deviation
increased by more than twenty times. The increase in standard deviation indicated that there was
greater variation in the modified dataset as compared to the original dataset. From the analysis,
the median was least affected by the addition of the supercar to the data. The median is
,therefore, best in representing the data as it is more central as compared to the mean and is least
affected by outliers. In the original dataset, the standard deviation was lower than the mean while
in the modified dataset the standard deviation was greater than the mean. The standard deviation
can, therefore, be used to detect outliers by comparing the value of standard deviation with the
mean. If standard deviation is greater than the mean then there is a high likelihood that the data
has outliers.
Normal Distributions are the most common distributions in statistics. If a random
variable x is normally distributed with a mean ? and a standard deviation ?. x
~N(?, ?) ; x ~N(0, 1) Normal distributions are known as “bell-shaped curve”
To find the probabilities of normal distributions you need to convert the x values
to a standard normal z-curve.
The equation of the z – score; ?? =
??? ??
??
You need to rewrite the probabilities in the less than form to use the function in
EXCEL. We will use Excel to find Normal Probabilities. The probabilities do need
to be in the less than form to use Excel. This is very important.
The Less Than or Equal To Form is not as important because normal probabilities
are continuous not discrete. But the probabilities do need to be in the less than
form to use Excel.
Here are some common Normal Probabilities and how they would get re-written
to calculate in the less than form, to use Excel.
• P( x = j)
• P( x ? j)
• P( x ? j) same as 1 – P(x 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 or False. Then
close the parenthesis ) and hit Enter.
Type in a True when you have a less than probability and type in a False when you
have an equals probability. This example has an “ >“ sign so we will use a True.
The probability that the next 5 cars sampled with be more than $25,000 is
66.15%.
Remember: Once you hit “Enter” the answer returns a decimal. You need to
convert it to a percent.
4) For the next 5 cars that are sampled, what is the probability that they will be
between $24,000 and $25,000?
Because of the word “between”, we want to find this probability P(24000