Question

Someone claims that the breaking strength of their climbing rope is 2,000 psi, with a standard deviation of 10 psi. We think the actual amount is lower than that and want to run the test at an alpha level of 5%. What would our sample size need to be if we want to reject the null hypothesis if the sample mean is at or below 1,997.2956?

Answer 1

Answer:

The sample size must be greater than 37 if we want to reject the null hypothesis.

Step-by-step explanation:

We are given that someone claims that the breaking strength of their climbing rope is 2,000 psi, with a standard deviation of 10 psi.

Also, we are given a level of significance of 5%.

Let $\mu$ = mean breaking strength of their climbing rope

SO, Null Hypothesis, $H_0$ : $\mu$ = 2,000 psi       {means that the mean breaking strength of their climbing rope is 2,000 psi}

Alternate Hypothesis, $H_A$ : $\mu$ < 2,000 psi      {means that the mean breaking strength of their climbing rope is lower than 2,000 psi}

Now, the test statistics that we will use here is One-sample z-test statistics as we know about population standard deviation;

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

where, $\bar X$ = ample mean strength = 1,997.2956 psi

            $\sigma$ = population standard devaition = 10 psi

            n = sample size

Now, at the 5% level of significance, the z table gives a critical value of -1.645 for the left-tailed test.

So, to reject our null hypothesis our test statistics must be less than -1.645 as only then we have sufficient evidence to reject our null hypothesis.

SO,  T.S. < -1.645   {then reject null hypothesis}

         $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } } < -1.645$

         $\frac{1,997.2956-2,000}{\frac{10}{\sqrt{n} } } < -1.645$

         $(\frac{1,997.2956-2,000}{10}) \times {\sqrt{n} } } < -1.645$

          $-0.27044 \times \sqrt{n}< -1.645$

               $\sqrt{n}> \frac{-1.645}{-0.27044}$

                 $\sqrt{n}>6.083$

                  n > 36.99 ≈ 37.

SO, the sample size must be greater than 37 if we want to reject the null hypothesis.