Question

The diameter of cork of a Champagne bottle is supposed to be 1.5 cm. If the cork is either too large or too small, it will not fit in the bottle. The manufacturer measures the diameter in a random sample of 36 bottles and finds their mean diameter to be 1.4 cm with standard deviation of 0.5 cm. Is there evidence at 1% level that the true mean diameter has moved away from the target?

Answer 1

`t_(n-1,\alpha/2)=3.59114678`

Therefore we do not have sufficient evidence at
`1\%` level that the true mean diameter has moved away from the target

Explanation:

From the question we are told that:

Sample size
`n=36`

Mean diameter
`\=x=1.4`

Standard deviation
`\sigma=0.5cm`

Null hypothesis
`H_0 \mu=1.5`

Alternative hypothesis
`\mu \\eq 1.5`

Significance level
`1\%=0.001`

Generally the equation for test statistics is mathematically given by

`t=(\=x-\mu)/((s)/(√(n) ) )`

`t=(1.4-1.5)/((0.5)/(√(36) ) )`

`t=-1.2`

Therefore since this is a two tailed test

`t_(n-1,\alpha/2)`

Where

`n-1=36-1=>35`

`\alpha=/2=0.001/2=>0.0005`

From table

`t_(n-1,\alpha/2)=3.59114678`

Therefore we do not have sufficient evidence at
`1\%` level that the true mean diameter has moved away from the target