Question

A local tennis pro-shop strings tennis rackets at the tension(pounds per square inch) requested by the customer. Recently a customer made a claim that the pro-shop consistently strings rackets at lower tensions, on average, than requested. To support this claim, the customer asked the pro shop to string 11new rackets at 51psi. Upon receiving the rackets, the customer measured the tension of each and calculated the following summarystatistics:x=48 psi,s=3.5 psi. In order to conduct the test, the customer selected a significance level ofα=0.01. Interpret this value.

Answer 1

Given information

Hypothesized mean \(=51\) psi

The sample mean \(=48\)

Sample Standard deviation \(s=3.5\)

\(n=11\) rackets

Claim: The pro-shop consistently strings rackets at lower tensions on average than requested. that is: the mean is less than 51 psi.

Hypothesis

Null Hypothesis \(\mathrm{H}_{0}: \mu=51 \mathrm{psi}\)

Alternative Hypothesis \(\mathrm{H}_{\mathrm{a}}: \mu<51\)psi

Since population standard deviation is unknown, we will use one sample t-test

Test Statistic

\(t=\frac{\bar{\chi}-\mu}{\frac{S}{\sqrt{n}}}\)

\(t=\frac{48-51}{\frac{3.5}{\sqrt{11}}}=-2.843\)

t-critical \(=-2.763769\) (from excel =T.INV (0.01,10)\()\)

\(|t=-2.8 .43|>\mid t-\) critical \(=-2.763769 \mid\)

\(\mathrm{P}\) -value \(=0.0087292<0.01\)

Since P-value \(=0.0087292<0.01\)level of significance, so we reject the null hypothesis H0.

Conclusion:

Sufficient Evidence to conclude that the true mean tension of the rackets is less than 51 psi.