what is the proportion of students with a raw score less than 10? if information was collected from fellow students the rating on a 5 point scale on 5 different questions. given a normally distributed population with a mean of 13, a standard deviation of 3, and 50 as the number of cases. what is the percentage of students with a raw score less than 20?
what is the number of students with raw scores between 10 and 20?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores. For percentage, multiply by 100. For n, multiply by 50.
Given a normally distributed population with a mean (μ) of 7, a standard deviation (σ) of 3, and 50 as the
number of cases (N), answer the following questions:
4.1 What is the proportion of cases with a raw score greater than 5?
4.2 What is the percentage of cases with a raw score greater than 15?
4.3 What is the number of cases with raw scores between 5 and 15?
To find the proportion of students with a raw score less than 10, we need to calculate the z-score for 10 first. The z-score formula is given by:
z = (x - μ) / σ
where:
x is the raw score
μ is the mean
σ is the standard deviation
In this case, x = 10, μ = 13, and σ = 3.
z = (10 - 13) / 3
z = -3 / 3
z = -1
We can now look up the z-score in the standard normal distribution table to find the proportion of students with a z-score less than -1. From the table, we find that the area to the left of -1 is approximately 0.1587.
Therefore, the proportion of students with a raw score less than 10 is approximately 0.1587.
To find the percentage of students with a raw score less than 20, we can follow the same steps as above.
z = (20 - 13) / 3
z = 7 / 3
z = 2.33
Looking up the z-score of 2.33 in the standard normal distribution table, we find that the area to the left of 2.33 is approximately 0.9900.
Therefore, the percentage of students with a raw score less than 20 is approximately 99.00%.
To find the number of students with raw scores between 10 and 20, we can subtract the proportion of students with a raw score less than 10 from the proportion of students with a raw score less than 20.
Proportion of students with a raw score between 10 and 20 = Proportion of students with a raw score less than 20 - Proportion of students with a raw score less than 10
= 0.9900 - 0.1587
= 0.8313
Finally, we can multiply the proportion by the total number of students (50) to find the number of students with raw scores between 10 and 20.
Number of students with raw scores between 10 and 20 = Proportion of students with a raw score between 10 and 20 * Total number of students
= 0.8313 * 50
= 41.57
Since the number of students must be a whole number, we round to the nearest whole number.
Therefore, the number of students with raw scores between 10 and 20 is approximately 42.
To find the proportion of students with a raw score less than 10, we need to use the normal distribution and the information provided. Here's how we can calculate it step by step:
1. Determine the Z-score: The Z-score measures the number of standard deviations a particular raw score is away from the mean. We can calculate the Z-score using the formula: Z = (X - μ) / σ, where X is the raw score, μ is the mean, and σ is the standard deviation.
In this case, we have X = 10, μ = 13, and σ = 3. Plugging these values into the formula, we get: Z = (10 - 13) / 3 = -1.
2. Find the proportion: Once we have the Z-score, we can find the proportion of students with a raw score less than 10 using a standard normal distribution table or a statistical calculator.
For a Z-score of -1, the standard normal distribution table tells us that the proportion of students with a raw score less than 10 is approximately 0.1587 (or 15.87%).
To find the percentage, we can multiply this proportion by 100: 0.1587 * 100 = 15.87%. So, approximately 15.87% of students have a raw score less than 10.
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To calculate the percentage of students with a raw score less than 20, we follow the same steps as above:
1. Calculate the Z-score: X = 20, μ = 13, σ = 3. Using the formula, we get: Z = (20 - 13) / 3 = 2.33.
2. Determine the proportion: Using the standard normal distribution table or a statistical calculator, we find that the proportion of students with a raw score less than 20 (Z-score = 2.33) is approximately 0.9904 (or 99.04%).
Multiplying this proportion by 100, we get: 0.9904 * 100 = 99.04%. Thus, approximately 99.04% of students have a raw score less than 20.
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To find the number of students with raw scores between 10 and 20, we need to find the difference in the proportions of students with raw scores less than 20 and less than 10.
Since we already calculated the proportions, we have:
Proportion less than 20 = 0.9904
Proportion less than 10 = 0.1587
Subtracting the proportion less than 10 from the proportion less than 20: 0.9904 - 0.1587 = 0.8317.
To find the number of students, we multiply this proportion by the total number of cases (50): 0.8317 * 50 = 41.585.
Therefore, there are approximately 41.585 students with raw scores between 10 and 20.