Question

a sports statistician claim that the mean winning times for boston marathon women's open division champions is at least 2.68 hours. the mean winning time of a sample of 30 randomly selected boston marathon women's open division champions is 2.60 hours. assume the population standard deviation is 0.32 hours. at the 5% level of significance can you reject the claim? use the p-value approach.

Answer 1

Answer:

We conclude that the mean winning times for Boston marathon women's open division champions is at least 2.68 hours.

Step-by-step explanation:

We are given that a sports statistician claim that the mean winning times for Boston marathon women's open division champions is at least 2.68 hours.

The mean winning time of a sample of 30 randomly selected Boston marathon women's open division champions is 2.60 hours. assume the population standard deviation is 0.32 hours.

Let $\mu$ = mean winning times for Boston marathon women's open division champions.

So, Null Hypothesis, $H_0$ : $\mu\geq$ 2.68 hours      {means that the mean winning times for Boston marathon women's open division champions is at least 2.68 hours}

Alternate Hypothesis, $H_A$ : p < 2.68 hours      {means that the mean winning times for Boston marathon women's open division champions is less than 2.68 hours}

The test statistics that would be used here One-sample z test statistics as we know about the population standard deviation;

                        T.S. =  $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$  ~ N(0,1)

where, $\bar X$ = sample mean winning time = 2.60 hours

            $\sigma$ = population standard deviation = 0.32 hours

            n = sample of women's open division champions = 30

So, test statistics  =   $\frac{2.60-2.68}{\frac{0.32}{\sqrt{30} } }$

                               =  -1.37

The value of z test statistics is -1.37.

Now, P-value of the test statistics is given by following formula;

         P-value = P(Z < -1.37) = 1 - P(Z $\leq$ 1.37)

                       = 1 - 0.9147 = 0.0853

Since, P-value of the test statistics is more than the level of significance as 0.0853 > 0.05, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.

Therefore, we conclude that the mean winning times for Boston marathon women's open division champions is at least 2.68 hours.