For a normally distributed distribution of variable x, where the mean is 50, and standard deviation is 2.5, calculate: a) the percentile rank of x=45 b) the z-score of x=52.6 c) the percentile rank of x=58 d) the 29.12th percentile e) the 89.74th percentile f) the z-score of x=45 g) the percentile rank of x=49
Z = (x - μ)/SD, where μ = mean.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the percentile ranks relates to the Z scores.
a. 0.5000 - 0.4772 = 0.0228, 2.28%, 2.28th percentile
b. Ƶ = 1.04 = 0.3508
c. 0.5000 + 0.4993 = 0.9993, 99.93%, 99.93rd percentile
d.
e.
f.
g. 0.5000 – 0.1554 =0.3446, 34.46%, 34.46th percentile
a) To calculate the percentile rank of x=45, we need to use the Z-score formula. The Z-score formula is:
Z = (x - μ) / σ
Where:
Z = Z-score
x = Value of x
μ = Mean
σ = Standard deviation
In this case, x = 45, μ = 50, and σ = 2.5. Plugging in these values into the formula:
Z = (45 - 50) / 2.5
Z = -2
To find the percentile rank, we need to use a Z-table or a statistical calculator. The percentile rank corresponding to a Z-score of -2 is approximately 2.28%.
b) To calculate the Z-score of x=52.6, we can use the Z-score formula mentioned above. Plugging in the values:
Z = (52.6 - 50) / 2.5
Z = 1.04
c) To calculate the percentile rank of x=58, we again use the Z-score formula:
Z = (58 - 50) / 2.5
Z = 3.2
Using the Z-table or a statistical calculator, the percentile rank corresponding to a Z-score of 3.2 is approximately 99.87%.
d) To find the value corresponding to the 29.12th percentile, we need to use the Z-table or a statistical calculator. The Z-score corresponding to the 29.12th percentile is approximately -0.54.
Using the Z-score formula:
Z = -0.54
Plugging in the values into the formula:
x = Z * σ + μ
x = -0.54 * 2.5 + 50
x = 48.65
Therefore, the 29.12th percentile corresponds to a value of approximately 48.65.
e) To find the value corresponding to the 89.74th percentile, we again use the Z-table or a statistical calculator. The Z-score corresponding to the 89.74th percentile is approximately 1.25.
Using the Z-score formula:
Z = 1.25
Plugging in the values into the formula:
x = Z * σ + μ
x = 1.25 * 2.5 + 50
x = 53.13
Therefore, the 89.74th percentile corresponds to a value of approximately 53.13.
f) To calculate the Z-score of x=45, we can use the Z-score formula mentioned earlier. Plugging in the values:
Z = (45 - 50) / 2.5
Z = -2
g) To calculate the percentile rank of x=49, we can again use the Z-score formula:
Z = (49 - 50) / 2.5
Z = -0.4
Using the Z-table or a statistical calculator, the percentile rank corresponding to a Z-score of -0.4 is approximately 34.13%.
To calculate the various values in your question, we need to use the properties of a normal distribution.
1. Percentile Rank of x=45:
To find the percentile rank of a given value, we need to calculate the area to the left of that value on the normal distribution curve. We can use the Z-score formula.
First, find the Z-score of x=45 using the formula:
Z = (x - μ) / σ
Z = (45 - 50) / 2.5
Z = -2
Next, we can look up the area to the left of Z=-2 in the Z-table. This gives us a value of approximately 0.0228 or 2.28%. Therefore, the percentile rank of x=45 is approximately 2.28%.
2. Z-Score of x=52.6:
To find the Z-score of a given value using a normal distribution, we use the formula:
Z = (x - μ) / σ
Z = (52.6 - 50) / 2.5
Z = 1.04
So the Z-score of x=52.6 is approximately 1.04.
3. Percentile Rank of x=58:
To find the percentile rank of x=58, we first need to calculate its Z-score using the same formula as above:
Z = (x - μ) / σ
Z = (58 - 50) / 2.5
Z = 3.2
Looking up the area to the left of Z = 3.2 in the Z-table, we get an approximate value of 0.9993 or 99.93%. Therefore, the percentile rank of x=58 is approximately 99.93%.
4. The 29.12th Percentile:
To find the value corresponding to a given percentile, we will use the inverse Z-score formula.
First, find the Z-score corresponding to the 29.12th percentile by looking it up in the Z-table. The closest Z-score to 29.12% is -0.57.
Using the inverse Z-score formula, we have:
x = Z * σ + μ
x = -0.57 * 2.5 + 50
x = 48.57
Therefore, the value at the 29.12th percentile is approximately 48.57.
5. The 89.74th Percentile:
To find the value corresponding to the 89.74th percentile, we'll use the same method as above.
First, find the Z-score closest to 89.74% in the Z-table, which is approximately 1.26.
Using the inverse Z-score formula:
x = Z * σ + μ
x = 1.26 * 2.5 + 50
x = 53.15
Therefore, the value at the 89.74th percentile is approximately 53.15.
6. Z-Score of x=45:
We previously calculated the Z-score of x=45 above as -2.
Therefore, the Z-score of x=45 is -2.
7. Percentile Rank of x=49:
To find the percentile rank of x=49, we need to calculate the area to the left of that value on the normal distribution curve. We can use the Z-score formula.
First, find the Z-score of x=49 using the formula:
Z = (x - μ) / σ
Z = (49 - 50) / 2.5
Z = -0.4
Next, we can look up the area to the left of Z=-0.4 in the Z-table. This gives us a value of approximately 0.3446 or 34.46%. Therefore, the percentile rank of x=49 is approximately 34.46%.