Answer to Question #188376 in Statistics and Probability for Ryzhel

Question #188376

Use the seven-step method to solve the following.

In a plant nursery, the owner thinks that the lengths of seedlings in a box sprayed with a new kind of fertilizer has an average heights of 26 cm after three days and a standard deviation of 10 cm. One researcher randomly selected 80 such seedlings and calculated the mean height to be 20 cm and the standard deviation was 10 cm. Will you conduct a one-tailed or a two-tailed test? Proceed with the test using = 0.05.


1
Expert's answer
2021-05-07T10:37:38-0400

Given μ0=26,xˉ=20,s=10,n=18.\mu_0=26, \bar{x}=20, s=10, n=18.


The following null and alternative hypotheses need to be tested:

H0:μ=26H_0:\mu=26

H1:μ26H_1:\mu\not=26


This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.


Based on the information provided, the significance level is α=0.05,\alpha=0.05, and the critical value for a two-tailed test with degrees of freedom df=n1=181=17df=n-1=18-1=17 is tc=2.109816.t_c=2.109816.


The rejection region for this two-tailed test is R={t:t>2.109816}.R=\{t:|t|>2.109816\}.

  

The t-statistic is computed as follows:



t=xˉμ0s/n=202610/18=2.545584t=\dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}=\dfrac{20-26}{10/\sqrt{18}}=-2.545584


Since it is observed that t=2.545584>2.109816=tc,|t|=2.545584>2.109816=t_c, it is then concluded that the null hypothesis is rejected.


Therefore, there is enough evidence to claim that the population mean μ\mu is different than 26 cm,26\ cm, at the 0.050.05 significance level.


Using the P-value approach:


The p-value for two-tailed, df=17,α=0.05,t=2.545584df=17, \alpha=0.05, t=-2.545584 is p=0.020881,p=0.020881, and since p=0.020881<0.05=α,p=0.020881<0.05=\alpha, it is then concluded that the null hypothesis is rejected.


Therefore, there is enough evidence to claim that the population mean μ\mu is different than 26 cm,26\ cm, at the 0.050.05 significance level.


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