The police department of a major city has found that the average height of their 1,250 officers is 71 inches (μ = 71 in.) with σ = 2.3 inches.
There are two companies that sell the medication, Company A and Company B. Company A’s tablets have a mean of 5 mg of medication and Company B’s tablets also have a mean amount of 5 mg of medication. This information would lead you to believe that it would make no difference which company filled your prescription. But what if you were told that Company A’s tablets have a standard deviation of 1 mg and Company B’s a standard deviation of .001 mg.
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I would want the one with the lowest variability. SD is a measure of variability.
To find the answer to your question, we need to use the concept of probability and the normal distribution.
The information you provided gives us the mean height of the police officers (μ = 71 inches) and the standard deviation (σ = 2.3 inches). These are the parameters of a normal distribution.
In order to answer the question, we need to know what specifically we're looking for. Here are a few examples of questions we could answer using this information:
1. What is the probability that a randomly selected police officer from this department has a height between 68 and 74 inches?
To find this probability, we can use the properties of the normal distribution. We will first standardize the values of 68 and 74 inches by subtracting the mean (71 inches) and dividing by the standard deviation (2.3 inches). This gives us the z-scores.
z1 = (68 - 71) / 2.3 = -1.30
z2 = (74 - 71) / 2.3 = 1.30
We can then use a standard normal distribution table or a statistical calculator to find the probability corresponding to these z-scores. The probability of a randomly selected police officer having a height between 68 and 74 inches is the difference between these probabilities.
2. What is the probability that a randomly selected police officer from this department is taller than 75 inches?
Again, we need to standardize the value of 75 inches to find the corresponding z-score.
z = (75 - 71) / 2.3 = 1.74
We can then use a standard normal distribution table or a statistical calculator to find the probability corresponding to this z-score. This gives us the probability that a randomly selected police officer is taller than 75 inches.
3. What is the probability that the average height of a sample of 50 police officers is less than 70 inches?
To answer this question, we need to use the concept of the sampling distribution of the mean. We know that the mean of the sampling distribution is the same as the population mean (μ = 71 inches).
The standard deviation of the sampling distribution (also called the standard error) can be calculated using the formula σ / sqrt(n), where σ is the population standard deviation (2.3 inches) and n is the sample size (50).
standard error = 2.3 / sqrt(50)
We can then standardize the value of 70 inches by subtracting the mean (71 inches) and dividing by the standard error.
z = (70 - 71) / (2.3 / sqrt(50))
We can now use a standard normal distribution table or a statistical calculator to find the probability corresponding to this z-score. This gives us the probability that the average height of a sample of 50 police officers is less than 70 inches.
These are just a few examples of how you can use the given information to answer different types of questions. The specific question you have will determine the approach you need to take.