This is a word problem. "We know that for certain workers the mean wage is $5/hr with a standard deviation of $.50. We need to find out if a worker is chosen at random what is the probability that the workers' wage is between $4.50 and $5.50? Assume a normal distribution of wages." I know that 68% of the workers' wages are between the two amounts but how do I determine the probability??
If you know that 68% of the wages are between the standard deviation range, which is true for a normal distribution, then you already have the answer. That IS the probability.
Here is a nice little applet that will illustrate what drwls said
http://davidmlane.com/hyperstat/z_table.html
In the mean enter 5
in the Sd enter .5
click on "between" and will in 4.5 and 5.5
the shaded area is both the percentage and the probability
To determine the probability that a worker's wage is between $4.50 and $5.50, you can use the concept of standard deviations. Since the wages are normally distributed, you can apply the empirical rule, also known as the 68-95-99.7 rule.
According to the rule, approximately 68% of the data falls within one standard deviation of the mean. In this case, with a mean wage of $5/hr and a standard deviation of $0.50, we can conclude that around 68% of workers earn between $4.50/hr and $5.50/hr.
To calculate the probability within this range, you can use the cumulative distribution function (CDF) of the normal distribution. Since the CDF calculates the probability of a value being less than or equal to a certain point, we need to find the probability up to $5.50/hr and then subtract the probability up to $4.50/hr.
Using a standard normal distribution table or a statistical software, you can find the cumulative probability for $5.50/hr. Let's assume the probability is P(X <= $5.50) = 0.8413.
Similarly, find the cumulative probability for $4.50/hr. Assume the probability is P(X <= $4.50) = 0.3085.
Now, subtract the two probabilities to get the probability within the specified range:
P($4.50/hr < X < $5.50/hr) = P(X <= $5.50) - P(X <= $4.50)
= 0.8413 - 0.3085
= 0.5328
Therefore, the probability that a randomly chosen worker's wage is between $4.50 and $5.50 is approximately 0.5328, or 53.28%.
To determine the probability that a worker's wage is between $4.50 and $5.50, we can utilize the concept of the z-score. The z-score measures how many standard deviations away from the mean a particular value is.
In this case, we can calculate the z-scores for $4.50 and $5.50 by using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
Using the given information, the mean (μ) is $5/hr, and the standard deviation (σ) is $0.50.
For $4.50:
z = ($4.50 - $5) / $0.50 = -0.5 / $0.50 = -1
For $5.50:
z = ($5.50 - $5) / $0.50 = 0.5 / $0.50 = 1
Now that we have the z-scores, we can use a standard normal distribution table or a calculator to find the probability associated with these values.
Referencing a standard normal distribution table, the probability corresponding to a z-score of -1 is approximately 0.1587. The probability corresponding to a z-score of 1 is also approximately 0.8413.
To find the probability between $4.50 and $5.50, we subtract the cumulative probability associated with the lower z-score from the cumulative probability associated with the higher z-score:
Probability = 0.8413 - 0.1587 = 0.6826
Therefore, the probability that a worker's wage chosen at random falls between $4.50 and $5.50 is approximately 0.6826, which is equivalent to 68.26%.