Answer to Question #318375 in Statistics and Probability for Joshua

Question

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 height of 26cm. after three days and a standard deviation of 10cm. one researcher randomly selected 80 such seedlings and calculated the mean height to be 20 cm. and the standard deviation was 10cm. will you conduct a one tailed test or two- tailed test? proceed with the test with 95% confidence level

Answer 1

Since sample mean is much smaller than the claimed one, then it is appropriate to put forward next hypotheses:

$H_0:a=26$

$H_1:a<26$

So, it is one-tailed test

Test statistic: $T={\frac {(x-a)*\sqrt n} {\sigma}}$ , where x - sample mean, a - claimed mean, n - sample size, $\sigma$ - standard deviation. So, $T={\frac {(20-26)*\sqrt {80}} {10}}\approx-5.37$

Since sample size is big, then it is appropriate to use z-score as critical value, then

$P(Z<Cr)=1-\alpha=1-0.95=0.05\implies Cr=-1.64$

Since T<Cr, then we should conclude that there is enough statistical evidence to reject the null hypothesis and admit that the mean is smaller than 26cm