the average student sends about 200 text messages a day with the standard deviation of 25 texts.(show graph)
a.what is the probability that a student sends more than 250 messsages a day?
b.what is the probability that student sends less than 150 text messages a day?
c.what is the probability that a student sends between 180 and 240?
d.what is the probability that a student ssends exactly 250 text messages a day?
e.find the cutoff number fot the middle 60% of text messages sent per day?
f. find the top 25% of the number of text messages sent per day?
g.find the bottom 10% of text messages sent per day
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions for the Z scores.
For e, f, g, find the proportion first and use the Z score in the above equation.
Here is the graph:
http://en.wikipedia.org/wiki/Standard_deviation
To solve these questions, we will use the concept of the normal distribution. The normal distribution is a symmetric bell-shaped curve that represents a continuous probability distribution. In this case, we will assume that the number of text messages sent per day follows a normal distribution.
However, before we proceed with calculating probabilities, we need to convert the given information into a standard normal distribution. This can be done by calculating the z-scores using the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we want to find the probability for
- μ is the mean (average) of the distribution
- σ is the standard deviation of the distribution
With this formula, we can calculate the z-scores for the given values in order to find the probabilities.
a. Probability of sending more than 250 messages a day:
To find the probability of sending more than 250 messages, we need to find the area under the normal distribution curve to the right of 250.
First, calculate the z-score:
z = (250 - μ) / σ
Given that the average student sends about 200 messages a day (μ = 200) and the standard deviation is 25 (σ = 25), we can plug in these values to calculate the z-score:
z = (250 - 200) / 25 = 2
Next, we can use a standard normal distribution table or a statistical software to find the probability associated with a z-score of 2. This probability represents the area to the right of the z-score.
b. Probability of sending less than 150 messages a day:
To find the probability of sending less than 150 messages, we need to find the area under the normal distribution curve to the left of 150.
Using the same approach, we calculate the z-score:
z = (150 - μ) / σ
By plugging in the values, we can calculate the z-score:
z = (150 - 200) / 25 = -2
Using the standard normal distribution table or software, we can find the probability associated with a z-score of -2, representing the area to the left of the z-score.
c. Probability of sending between 180 and 240 messages a day:
To find the probability of sending between 180 and 240 messages, we need to find the area under the normal distribution between these two values.
First, we calculate the z-scores for each value:
z1 = (180 - μ) / σ
z2 = (240 - μ) / σ
Calculate the z-scores using the average and standard deviation:
z1 = (180 - 200) / 25 = -0.8
z2 = (240 - 200) / 25 = 1.6
Using the standard normal distribution table or software, we can find the probabilities associated with these z-scores. Then, we calculate the difference between the two probabilities to find the area between them.
d. Probability of sending exactly 250 messages a day:
Since the normal distribution is a continuous distribution, the probability of having an exact value is zero. Therefore, the probability of sending exactly 250 messages a day is negligible.
e. Cutoff number for the middle 60% of text messages sent per day:
To find the cutoff number for the middle 60%, we need to find the z-scores that correspond to the 20th and 80th percentiles of the normal distribution.
The z-score corresponding to the 20th percentile is found by looking up the value in the standard normal distribution table or by using software. Similarly, we can find the z-score corresponding to the 80th percentile.
Using the formula:
x = μ + z * σ
where z is the z-score and x is the cutoff number, we can find the cutoff numbers for the middle 60% of text messages sent per day.
f. Top 25% of the number of text messages sent per day:
To find the top 25%, we need to find the cutoff number that separates the top 25% from the rest of the distribution.
First, we need to find the z-score corresponding to the 75th percentile using the standard normal distribution table or software. Then, using the formula mentioned in the previous step, we can calculate the cutoff number.
g. Bottom 10% of text messages sent per day:
To find the bottom 10%, we need to find the cutoff number that separates the bottom 10% from the rest of the distribution.
Using the same approach as before, find the z-score corresponding to the 10th percentile. Then, use the formula to calculate the cutoff number for the bottom 10%.
Note that for all the above calculations involving the standard normal distribution, it can be helpful to use statistical software or online calculators to find the z-scores and probabilities accurately and efficiently.