Question

5. A scientist believes the concentration of radon gas in the air is greater that the established safe level of 4pci or less. The scientist tests the composition for 36 days finding an average concentration of 4.4 pci with a sample standard deviation of 1 pci. a) In testing the scientist's belief, write the appropriate hypotheses at 0.05 level of significance.

Answer 1

Complete Question

A scientist believes the concentration of radon gas in the air is greater that the established safe level of 4 pci or less. The scientist tests the composition for 36 days finding an average concentration of 4.4 pci with a sample standard deviation of 1 pci.

a) In testing the scientist's belief, write the appropriate hypotheses:

           $H_o:$                   Ha:

b) What decision should be made?

Answer:

a

The null hypothesis is  $H_o : \mu \le 4 \ pci$

The  alternative  hypothesis is $H_a : \mu > 4 \ pci$

b

There is sufficient evidence to conclude that the concentration of radon gas in the air is greater that the established safe level of 4pci or less

Step-by-step explanation:

From the question we are told that

   The population mean is  $\mu = 4 \ pci$

    The sample size $n = 36 \ days$

     The  sample  mean is  $\= x = 4.4 \ pci$

     The  standard deviation is  $\sigma = 1 \ pci$

       The  level  of significance is  $\alpha = 0.05$

The null hypothesis is  $H_o : \mu = 4 \ pci$

The  alternative  hypothesis is $H_a : \mu > 4 \ pci$

  Generally the test statistics is mathematically represented as

          $t = \frac{ \= x - \mu }{ \frac{ \sigma}{ \sqrt{n} } }$

=>      $t = \frac{ 4.4 - 4 }{ \frac{ 1}{ \sqrt{36} } }$

=>      $t = \frac{ 4.4 - 4 }{ \frac{ 1}{ \sqrt{36} } }$

=>      $t =2.4$

The  p-value  is mathematically represented as

      $p-value = P(Z > 2.4 )$

From the  z-table  

          $P(Z > 2.4 ) = 0.008$

    $p-value =0.008$

So from this obtained value  we see that

     $p-value < \alpha$ so we reject the null hypothesis

Hence we can conclude that there is sufficient evidence to conclude that the concentration of radon gas in the air is greater that the established safe level of 4pci or less