An article in a Health Magazine suggested getting a dog to encourage more regular walking. A researcher for the article found that the distribution of time spent walking dogs is approximately normally distributed with a mean of 38 minutes per day. However he also found that 13% of dog owners walk their dogs less than 20 minutes per day. What is the standard deviation of the time, in minutes, dog owners walk their dog per day?
I don't know how to find standard deviation. I tried to use the z score formula to solve for SD but there are too many unknown variables. There must be some calculator function (because doing it by hand is hard and takes forever) but I don't know it. Please help, thanks!
17.48
Well, calculating the standard deviation can be a bit tedious and time-consuming, just like walking a dog for less than 20 minutes a day. But don't worry, I'm here to help!
To find the standard deviation, we'll need to use some statistics magic involving the z-score and the cumulative standard normal distribution.
First, let's find the z-score using the formula:
z = (x - μ) / σ
where z is the z-score, x is the observed value, μ is the mean, and σ is the standard deviation.
Given that 13% of dog owners walk their dogs less than 20 minutes per day, we can say that this is the z-score.
Using the cumulative standard normal distribution table or calculator, we can find the corresponding z-score for a cumulative probability of 0.13. In this case, it's approximately -1.04.
Now, setting the z-score equal to -1.04, we can rearrange the formula to solve for the standard deviation (σ):
-1.04 = (20 - 38) / σ
Simplifying it:
-1.04σ = -18
Dividing by -1.04:
σ ≈ 17.31
So, the standard deviation is approximately 17.31 minutes.
Now you know the standard deviation, which is just one step closer to walking the dog (or understanding statistical concepts)! Keep up the good work, and remember to bring a pooper scooper along on your walks!
To find the standard deviation of the time dog owners walk their dogs per day, we need to use a concept called the z-score. However, you don't need to solve for the standard deviation by using the z-score formula directly. Instead, we can use the properties of the normal distribution to solve this problem.
Given that the distribution of time spent walking dogs is approximately normally distributed with a mean of 38 minutes per day, and we know that 13% of dog owners walk their dogs less than 20 minutes per day, we can use a standard normal distribution table to find the z-score corresponding to this percentile.
First, let's find the z-score for the 13th percentile, which is the same as walking less than 20 minutes per day. In a standard normal distribution, the z-score corresponding to a given percentile can be found using a z-table or calculator.
Using a z-table, the closest percentile value we can find to 0.13 is 0.1292. The corresponding z-score is approximately -1.08. This means that the value 20 minutes is -1.08 standard deviations away from the mean.
Now, we can calculate the standard deviation (SD) using the z-score formula:
z = (X - μ) / σ
Rearranging the formula to solve for the standard deviation (σ), we get:
σ = (X - μ) / z
Substituting the known values into the formula:
σ = (20 - 38) / -1.08
Calculating this, we find:
σ ≈ 17.8
Therefore, the standard deviation of the time dog owners walk their dogs per day is approximately 17.8 minutes.
If you prefer to use a calculator, you can use statistical software like Microsoft Excel or online calculators (such as Symbolab, Stat Trek, or Desmos) that have built-in functions for calculating z-scores and standard deviations from percentiles. Just enter the values and follow the instructions provided by the calculator.
you have the mean
go to the z-score table and find 0.1300
... this will tell you the number of standard deviations
divide the number into 20 min to find the SD