math question

Normal Distributions are the most common distributions in statistics. If a random variable 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;

𝑧 = 𝑥 − 𝜇

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 (<). Less Than OR Less Than and Equal To is not as important because normal probabilities are continuous not discrete. 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) same as P( X < j) • P( X ≥ j) same as 1 – P(X < j ) • P(j < X < k) = P(X < k) – P(X < j) • Expected Value = µ (Mean) • Standard deviation = σ (SD)

To find Normal Probabilities we will use the =NORM.DIST( ) function. The Central Limit Theorem states that given any distribution with a mean μ and a standard deviation of σ, the sample mean will approach a normal distribution as the sample size, n, increases. The new mean of the sample mean will equal the old mean; new μ = old μ and the new standard deviation of the sample mean (this is also called standard error) will be written as;

𝒏𝒆𝒘 𝒔𝒅 (or standard error) = 𝜎

√𝑛

Let’s use our Car Price Data from Week 1 and calculate 4 different probabilities.

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 1. Using our data, we believe that the cost of the type of car we calculated is normally distributed with a mean of $25,650 and a SD of $3,488.47. 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 less than $24,000? The probability is already in the less than form, P(�̅� < 24,000), so we do not need to do additional work in Excel to find the probability. We also notice that the new sample size is n = 5. The mean will stay the same, but we will need to calculate a new SD. We will apply the Central Limit Theorem to do this. Remember you need to put in the “=” sign and then we will click on the cell that contains the old SD, and will hit the “ / “ sign and then use the SQRT( ) function and put 5 within the parentheses because the new sample size is 5.

𝒏𝒆𝒘 𝒔𝒅 = 𝜎

√𝑛 =

3488.47

√5 = 1560.09

Next, we want to find this probability P(�̅� < 24,000) and we will use the NORM.DIST() function in Excel to do this. P(�̅� < 24000) = 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 ) and hit Enter. ALWAYS type in a True for continuous probability functions (the normal distribution is continuous). This example has an “<“ sign so we will use a True.

The probability that the sample mean for the new sample of 5 cars is below $24,000 is 14.51%. 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.

2. 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 higher than $25,000? Because of the words “higher than”, we want to find this probability P(�̅� > 25,000). 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(�̅� > 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 < �̅�< 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 < �̅� < 25000) = P(�̅� < 25000) – P(�̅� < 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.