Answer:
The standard deviation is 1.5
Explanation:
Use the z-score formula to find the standard deviation.
[tex]z = \frac{x - \mu}{ \sigma} [/tex]
Using the empirical rule 95% falls within 2 standard deviations of the mean.
Using the upper limit, we have x=89 and
[tex] \mu = 86[/tex]
We substitute and solve for the standard deviation.
[tex]2= \frac{89- 86}{ \sigma} [/tex]
This means
[tex]\sigma= \frac{89- 86}{ 2} = \frac{3}{2} [/tex]
[tex]\sigma = 1.5[/tex]
The standard deviation of the distribution in the Dominican Republic in August is 1.53
The distribution represents a normal distribution
The mean temperature is:
[tex]\mu = 86^0F[/tex]
Approximately 95% of all daily high temperatures are between 83°F and 89°F
That is:
[tex]X_1 = 83^oF\\\\X_2 = 89^0F[/tex]
For 95% confidence interval, the z-value = 1.960
The equation showing the relationship between the z-value, the mean μ, the value of X, and the standard deviation σ is shown below:
[tex]z = \frac{X_2-\mu}{\sigma}[/tex]
Substitute X₂ = 89, μ = 86, and z = 1.960 into the equation above to solve for σ
[tex]1.96 = \frac{89-86}{\sigma} \\\\1.96 \sigma = 3\\\\\sigma = \frac{3}{1.96} \\\\\sigma = 1.53[/tex]
The standard deviation of the distribution is 1.53
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