A scale measuring prejudice has been administered to a large sample of respondents. The distribution of scores is approximately normal with a mean of 31 and a standard deviation of 5. What percentage of sample had scores
1) Below 20
2) Above 40
3) Above 20
4) Below 40
5) Below 25
6) Above 25
7) Above 34
8) Below 34
11
To calculate the percentage of respondents with scores below or above a certain value, you can use the standard normal distribution table. However, since the given distribution is approximately normal, we can still make an estimation using the standard normal distribution.
1) Percentage of sample with scores below 20:
To calculate this, we need to standardize the score using the formula:
z = (X - μ) / σ
where X is the score, μ is the mean, and σ is the standard deviation.
z = (20 - 31) / 5
z = -11 / 5
z = -2.2
Looking up the z-value -2.2 in the standard normal distribution table, we find that the percentage is approximately 1.34%.
2) Percentage of sample with scores above 40:
Again, we need to standardize the score.
z = (40 - 31) / 5
z = 9 / 5
z = 1.8
Looking up the z-value 1.8 in the standard normal distribution table, we find that the percentage is approximately 96.04%.
3) Percentage of sample with scores above 20:
Using the same process, we find the z-value:
z = (20 - 31) / 5
z = -11 / 5
z = -2.2
Since we want the percentage above 20, we need to find the complement of the percentage below 20. Subtracting this from 100%, we get:
100% - 1.34% = 98.66%
Approximately, 98.66% of the sample had scores above 20.
4) Percentage of sample with scores below 40:
Using the same process:
z = (40 - 31) / 5
z = 9 / 5
z = 1.8
Since we want the percentage below 40, we can directly use the z-value found previously:
95.96%
Approximately, 95.96% of the sample had scores below 40.
5) Percentage of sample with scores below 25:
Following the same steps:
z = (25 - 31) / 5
z = -6 / 5
z = -1.2
Looking up the z-value -1.2, we find that the percentage is approximately 11.92%.
6) Percentage of sample with scores above 25:
Again, using the same approach:
z = (25 - 31) / 5
z = -6 / 5
z = -1.2
To find the percentage above 25, we subtract this from 100%:
100% - 11.92% = 88.08%
Approximately, 88.08% of the sample had scores above 25.
7) Percentage of sample with scores above 34:
Using the same procedure:
z = (34 - 31) / 5
z = 3 / 5
z = 0.6
Looking up the z-value 0.6, we find that the percentage is approximately 72.16%.
8) Percentage of sample with scores below 34:
Applying the previous result, we subtract from 100%:
100% - 72.16% = 27.84%
Approximately, 27.84% of the sample had scores below 34.
To calculate the percentage of scores for the given scenarios, we will need to use the z-score formula and the standard normal distribution table (also known as the Z-table).
The z-score formula is:
z = (x - μ) / σ
where:
- x is the given value
- μ is the mean of the distribution
- σ is the standard deviation
Scenarios:
1) To find the percentage of scores below 20:
- Calculate the z-score: z = (20 - 31) / 5 = -2.2
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of -2.2 is approximately 1.62%.
Therefore, approximately 1.62% of the sample had scores below 20.
2) To find the percentage of scores above 40:
- Calculate the z-score: z = (40 - 31) / 5 = 1.8
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of 1.8 is approximately 96.78%.
Therefore, approximately 96.78% of the sample had scores above 40.
3) To find the percentage of scores above 20:
- Calculate the z-score: z = (20 - 31) / 5 = -2.2
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of -2.2 is approximately 1.62%.
Since we want the percentage above 20, we subtract the above percentage from 100%: 100% - 1.62% = 98.38%.
Therefore, approximately 98.38% of the sample had scores above 20.
4) To find the percentage of scores below 40:
- Calculate the z-score: z = (40 - 31) / 5 = 1.8
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of 1.8 is approximately 96.78%.
Since we want the percentage below 40, we subtract the above percentage from 100%: 100% - 96.78% = 3.22%.
Therefore, approximately 3.22% of the sample had scores below 40.
5) To find the percentage of scores below 25:
- Calculate the z-score: z = (25 - 31) / 5 = -1.2
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of -1.2 is approximately 11.12%.
Therefore, approximately 11.12% of the sample had scores below 25.
6) To find the percentage of scores above 25:
- Calculate the z-score: z = (25 - 31) / 5 = -1.2
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of -1.2 is approximately 11.12%.
Since we want the percentage above 25, we subtract the above percentage from 100%: 100% - 11.12% = 88.88%.
Therefore, approximately 88.88% of the sample had scores above 25.
7) To find the percentage of scores above 34:
- Calculate the z-score: z = (34 - 31) / 5 = 0.6
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of 0.6 is approximately 72.14%.
Since we want the percentage above 34, we subtract the above percentage from 100%: 100% - 72.14% = 27.86%.
Therefore, approximately 27.86% of the sample had scores above 34.
8) To find the percentage of scores below 34:
- Calculate the z-score: z = (34 - 31) / 5 = 0.6
- Look up the z-score in the standard normal distribution table to find the corresponding percentage. The table will tell you that the percentage for the z-score of 0.6 is approximately 72.14%.
Therefore, approximately 72.14% of the sample had scores below 34.
Note that these calculations are based on the assumption that the distribution of scores is approximately normal.