A business uses a 7-point scale about satisfaction with the services it provides to clients. The ratings are normally distributed with a mean of 4.8 and a standard deviation of .5. What percentage of clients rate their satisfaction
1) Above 5?
2) Above 6?
3) Above 4?
4) Below 4?
(*Write down just a number for each question with 2 decimal points like 99.99 without %. Include 0 before the decimal point like 0.01.)
5
5
5
To calculate the percentages for each question, we can use the Z-score formula:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we want to find the percentage for
μ is the mean
σ is the standard deviation
1) To find the percentage of clients who rate their satisfaction above 5, we need to find the Z-score for a value of 5.
Z = (5 - 4.8) / 0.5
= 0.4 / 0.5
= 0.8
Using a Z-table or calculator, we can find that the percentage of clients rating their satisfaction above 5 is approximately 21.95%.
2) To find the percentage of clients who rate their satisfaction above 6, we need to find the Z-score for a value of 6.
Z = (6 - 4.8) / 0.5
= 1.2 / 0.5
= 2.4
Using a Z-table or calculator, we can find that the percentage of clients rating their satisfaction above 6 is approximately 0.39%.
3) To find the percentage of clients who rate their satisfaction above 4, we need to find the Z-score for a value of 4.
Z = (4 - 4.8) / 0.5
= -0.8 / 0.5
= -1.6
Using a Z-table or calculator, we can find that the percentage of clients rating their satisfaction above 4 is approximately 92.15%.
4) To find the percentage of clients who rate their satisfaction below 4, we need to find the Z-score for a value of 4.
Z = (4 - 4.8) / 0.5
= -0.8 / 0.5
= -1.6
Using a Z-table or calculator, we can find that the percentage of clients rating their satisfaction below 4 is approximately 7.85%.
So, the answers to each question are as follows:
1) Above 5: 21.95%
2) Above 6: 0.39%
3) Above 4: 92.15%
4) Below 4: 7.85%
To find the percentages for each scenario, we need to use the standard normal distribution, also known as the Z-score.
The Z-score formula is:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Given value
μ = Mean
σ = Standard deviation
1) To find the percentage of clients who rate their satisfaction above 5, we first need to calculate the Z-score for 5.
Z = (5 - 4.8) / 0.5
Z = 0.4 / 0.5
Z = 0.8
Using a standard normal distribution table (or a calculator), we can find the percentage corresponding to a Z-score of 0.8. In this case, it is approximately 21.42%. Therefore, 21.42% of clients rate their satisfaction above 5.
2) To find the percentage of clients who rate their satisfaction above 6, we need to calculate the Z-score for 6.
Z = (6 - 4.8) / 0.5
Z = 1.2 / 0.5
Z = 2.4
Using the standard normal distribution table, we find that the percentage corresponding to a Z-score of 2.4 is approximately 99.73%. Therefore, 99.73% of clients rate their satisfaction above 6.
3) To find the percentage of clients who rate their satisfaction above 4, we need to calculate the Z-score for 4.
Z = (4 - 4.8) / 0.5
Z = -0.8 / 0.5
Z = -1.6
Using the standard normal distribution table, we find that the percentage corresponding to a Z-score of -1.6 is approximately 6.68%. However, we want to know the percentage above 4, not below. So we subtract this percentage from 100% to get the percentage above 4.
100% - 6.68% = 93.32%
Therefore, 93.32% of clients rate their satisfaction above 4.
4) To find the percentage of clients who rate their satisfaction below 4, we can subtract the percentage above 4 from 100%.
100% - 93.32% = 6.68%
Therefore, 6.68% of clients rate their satisfaction below 4.