The amount of Jen's monthly phone bill is normally distributed with a mean of $55 and a standard deviation of $12. Calculate the z score for each scenario.
What percentage of her phone bills are above 79?
What percentage of her phone bills are between $31 and $79.
What would be the z score and monthly bill of somone in the top 5% of monthly phone bills?
To calculate the z-scores for these scenarios, we can use the formula:
Z = (X - μ) / σ
Where:
Z = z-score
X = value of interest
μ = mean
σ = standard deviation
Let's calculate the z-scores for each scenario:
1. What percentage of her phone bills are above $79?
Z = (79 - 55) / 12 = 2
To find the percentage of phone bills above $79, we need to find the area under the normal distribution curve to the right of the z-score of 2. We can refer to the z-table or use a calculator to find this area. From the standard normal distribution table, we find that the percentage is approximately 2.28%.
2. What percentage of her phone bills are between $31 and $79?
Z1 = (31 - 55) / 12 = -2
Z2 = (79 - 55) / 12 = 2
To find the percentage of phone bills between $31 and $79, we need to find the area under the normal distribution curve between the z-scores of -2 and 2. Again, we can refer to the z-table or use a calculator to find this area. From the standard normal distribution table, we find that the percentage is approximately 95.44%.
3. What would be the z-score and monthly bill of someone in the top 5% of monthly phone bills?
To find the z-score corresponding to the top 5% of phone bills, we need to find the z-score that corresponds to an area of 0.05 in the right tail of the normal distribution curve. From the standard normal distribution table, the z-score that corresponds to an area of 0.05 in the right tail is approximately 1.645.
Now we can use the z-score formula to find the monthly bill corresponding to this z-score:
Z = (X - μ) / σ
1.645 = (X - 55) / 12
Solving for X, we have:
1.645(12) = X - 55
19.74 + 55 = X
X ≈ $74.74
Therefore, someone in the top 5% of monthly phone bills would have a monthly bill of approximately $74.74, with a corresponding z-score of approximately 1.645.
To calculate the z-scores for each scenario, we will use the formula:
z = (x - μ) / σ
where:
z is the z-score
x is the value we want to calculate the z-score for
μ is the mean of the distribution
σ is the standard deviation of the distribution
Now let's calculate the z-scores for each scenario:
1. What percentage of her phone bills are above $79?
To calculate this, we need to find the z-score for $79 using the formula:
z = ($79 - $55) / $12
z = 24 / 12
z = 2
Now, we need to find the area (percentage) under the normal distribution curve to the right of the z = 2. This can be done by referring to the z-table or using a statistical calculator. The percentage will provide us with the answer.
2. What percentage of her phone bills are between $31 and $79?
Similar to the previous scenario, we need to find the z-scores for both $31 and $79 and calculate the area (percentage) between these two z-scores. The formula for each z-score will be:
z1 = ($31 - $55) / $12
z2 = ($79 - $55) / $12
Again, we can use the z-table or a statistical calculator to find the area (percentage) under the normal distribution curve between z1 and z2.
3. What would be the z-score and monthly bill of someone in the top 5% of monthly phone bills?
To find the z-score in the top 5% of the distribution, we need to find the z-score corresponding to the area of 0.95 (1 - 0.05) under the normal distribution curve. We can find it using the z-table or a statistical calculator.
Once we have the z-score, we can calculate the monthly bill using the formula:
monthly bill = (z * standard deviation) + mean
By substituting the calculated z-score, standard deviation ($12), and mean ($55) into the formula, we can find the monthly bill amount for someone in the top 5% of monthly phone bills.
Remember, z-scores help us to standardize values and understand their relative positions within a distribution. By comparing z-scores, we can find the probabilities and percentiles associated with different values in a normal distribution.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for each Z score. For last question, use the table to find .05 and insert its Z score in the equation above.