Question #348920

1. A researcher estimates that the average height of the buildings of 30 or more stories in a large city is at least 700 feet. A random sample of 10 buildings is selected, and the heights in feet are shown. At  = 0.025, is there enough evidence to reject the claim? 485 511 841 725 615 520 535 635 616 582


1
Expert's answer
2022-06-08T16:15:05-0400
xˉ=110(485+511+841+725+615\bar{x}=\dfrac{1}{10}(485 +511+ 841+ 725+ 615

+520+535+635+616+582)=606.5+520 +535 +635 +616+ 582)=606.5

s2=i(xixˉ)2n1=1101((485606.5)2s^2=\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}=\dfrac{1}{10-1}((485-606.5)^2

+(511606.5)2+(841606.5)2+(511-606.5)^2+(841-606.5)^2

+(725606.5)2+(615606.5)2+(725-606.5)^2+(615-606.5)^2

+(520606.5)2+(535606.5)2+(520-606.5)^2+(535-606.5)^2

+(635606.5)2+(616606.5)2+(635-606.5)^2+(616-606.5)^2

+(582606.5)2)=107084.59+(582-606.5)^2)=\dfrac{107084.5}{9}

s=s2=107084.59109.08s=\sqrt{s^2}=\sqrt{\dfrac{107084.5}{9}}\approx109.08

The parameter is average content of fruit concentrate per bottle.

The following null and alternative hypotheses need to be tested:

H0:μ700H_0:\mu\ge700

H1:μ<700H_1:\mu<700

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

Based on the information provided, the significance level is α=0.025,\alpha = 0.025, df=n1=9df=n-1=9 and the critical value for a left-tailed test is tc=2.262156.t_c =-2.262156.

The rejection region for this left-tailed test is R={t:t<2.262156}.R = \{t:t<-2.262156\}.

The t-statistic is computed as follows:


t=xˉμs/n=606.5700109.08/10=2.7106t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{606.5-700}{109.08/\sqrt{10}}=-2.7106


Since it is observed that t=2.7106<2.262156=tc,t=-2.7106<-2.262156=t_c, it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for left-tailed, df=9df=9 degrees of freedom, t=2.7106t=-2.7106 is p=0.011987,p=0.011987, and since p=0.011987<0.025=α,p=0.011987<0.025=\alpha, it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean μ\mu

is less than 700, at the α=0.025\alpha = 0.025 significance level.


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