Question

Georgianna claims that in a small city renowned for its music school, the average child takes at least 5 years of piano lessons. We have a random sample of 20 children from the city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years. Required:Explicitly state and check all conditions necessary for inference on these data.

Answer 1

Answer:

The  condition  are

           The  Null hypothesis is  $H_o : \mu = 5$

           The  Alternative hypothesis is  $H_a : \mu < 5$

The  check revealed that

             There is sufficient evidence to support the claim that in a small city renowned for its music school, the average child takes at least 5 years of piano lessons

Step-by-step explanation:

From the question we are told that

     The  population mean is  $\mu = 5 \ year$

      The sample size is  n =  20

      The sample mean is  $\= x = 4.6 \ years$

       The  standard deviation is $\sigma = 2.2 \ years$

   The  Null hypothesis is  $H_o : \mu = 5$

   The  Alternative hypothesis is  $H_a : \mu < 5$

So i will be making use of  $\alpha = 0.05$ level of significance to test this claim

    The critical value of  $\alpha$ from the normal distribution table is  $Z_\alpha = 1.645$

 

Generally the test statistics is mathematically evaluated as

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

substituting values

                 $t = \frac{ 4.6 - 5}{ \frac{2.2}{\sqrt{20} } }$

                $t = -0.8131$

Looking at the value of  t and $Z_{\alpha }$ we see that  $t < Z_{\alpha }$ so we fail to reject the null hypothesis  

  This implies that there is sufficient evidence to support the claim that in a small city renowned for its music school, the average child takes at least 5 years of piano lessons.