Question

migraine headaches a study found that the mean number of days per month that people who suffer from migraine headaches is 15.7 days. assume that the variable is approximately normally distributed and the standard deviation is 2 days. find the probability that if a random sample of 50 people who suffer from migraine headaches is selected, the mean of the sample will be between 15 and 16 days. use * the standard normal distribution table. round intermediate value calculations to two decimal places, and the final answer to at least four decimal places.

p(15<\overline {x}<16)=\square

Explanation:

Step1: Calculate the standard error

The formula for the standard error (\(\sigma_{\bar{x}}\)) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma = 2\) (standard deviation) and \(n = 50\) (sample size).

$$ \sigma_{\bar{x}}=\frac{2}{\sqrt{50}}\approx\frac{2}{7.0711}\approx0.2828 $$

Step2: Calculate the \(z\)-scores

The \(z\)-score formula is \(z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}\)
For \(\bar{x} = 15\):

$$ z_1=\frac{15 - 15.7}{0.2828}=\frac{- 0.7}{0.2828}\approx - 2.47 $$

For \(\bar{x}=16\):

$$ z_2=\frac{16 - 15.7}{0.2828}=\frac{0.3}{0.2828}\approx1.06 $$

Step3: Find the probabilities

Using the standard normal distribution table:
\(P(Z < - 2.47)=0.0068\) and \(P(Z < 1.06)=0.8554\)

Step4: Calculate the required probability

$$ P(15<\bar{X}<16)=P(Z < 1.06)-P(Z < - 2.47)=0.8554 - 0.0068=0.8486\approx0.34\quad(\text{rounded to two decimal places}) $$

Answer:

\(0.34\)