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

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

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 = (\= x - \mu)/( (\sigma )/(√(n) ) )`

substituting values

`t = ( 4.6 - 5)/( (2.2)/(√(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.