What is the probability of 59 if the mean is 60 and the standard deviation is 7?
If you have a discrete distribution, i.e. the values are integers, then the probability of 59 would be the equivalent of the group 58.5 to 59.5 in a continuous distribution.
58.5 corresponds to z1=((μ-58.5)/σ) and 59.5 to z2=((μ-59.5)/σ).
Assuming the distribution is normal, look up the z-tables to find the one-tail probabilities of z1 and z2. The difference between z1 and z2 will be the probability of getting between 58.5 and 59.5.
The average commute time via train from the Chicago O'Hare Airport to downtown is 60 minutes with a standard deviation of 15 minutes. Assume that the commute times are normally distributed. What proportion of commutes would be:
longer than 80 minutes?
less than 50 minutes?
between 45 and 75 minutes?
Quantitative Research - Tina, Saturday, May 28, 2011 at 7:52pm
The average commute time via train from the Chicago O'Hare Airport to downtown is 60 minutes with a standard deviation of 15 minutes. Assume that the commute times are normally distributed. What proportion of commutes would be:
longer than 80 minutes?
less than 50 minutes?
between 45 and 75 minutes?
To calculate the probability of a specific value in a normally distributed data set, we need to standardize the value using the mean and standard deviation.
The first step is to calculate the z-score, which represents how many standard deviations away a specific value is from the mean. The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
In this case, we want to calculate the z-score for the value 59, with a mean of 60 and a standard deviation of 7:
z = (59 - 60) / 7 = -0.142857
The next step is to use a z-table (also known as a standard normal distribution table) to find the corresponding probability. The z-table provides the percentage of data below a given z-score.
Since the z-table typically provides probabilities for positive z-scores, we need to find the probability for the absolute value of the z-score. In this case, we would look up the z-score 0.142857 in the z-table.
The z-table shows that the probability corresponding to a z-score of 0.142857 (or -0.142857) is 0.5564. This means that the probability of getting a value of 59 in a normally distributed data set with a mean of 60 and a standard deviation of 7 is approximately 0.5564, or 55.64%.
Note that the z-table may have slight variations depending on the specific table being used, but the general process remains the same.