Question

The International Tennis Federation (ITF) requires that tennis balls have an average diameter of 6.7 centimetres. Tennis balls being produced by one manufacturer are tested to see if they fail to meet the ITF standard. A random sample of 9 tennis balls had a mean diameter of 6.55 cm, with standard deviation 0.12 cm. Assume tennis ball diameter is normally distributed.
Enter your responses to all parts in the text box below.  

a) Choose the correct hypotheses to test the research hypothesis that the manufacturer's tennis balls fail to meet the ITF standard.

b) Perform the hypothesis test.
i.e. Calculate then state the test statistic value, the P-value and your conclusion in the context of the question. Use α = 0.05.

c) Calculate then give a 95% confidence interval for the true mean diameter and interpret your interval.

d) Comment on the hypothesis test conclusion from part b) in relation to the confidence interval found in part c).

Answer 1

a)-

ITF requires that tennis balls have an average diametre of 6.7 cms. So, we have to test that average diametre of balls is equal to 6.7 cms or not. For this problem, we can use one sample t test for significance of mean as the sample size(9) is less than 30.

b)-

A random sample of 9 balls had a mean diametre of 6.55 cm with standard deviation of 0.12 cm.

So, = 6.55, s = 0.12, n =9

Hypothesis:

Null hypothesis- H₀ : i.e. Average diametre of balls is equal to 6.7 cms.

Alternative hypothesis - H₁ : i.e.Average diametre of balls is not equal to 6.7 cms.

Test statistic -

Test criterion -

reject H₀ if p<0.05 otherwise accept H₀ at 5% of level of significance.

Calculation -

P-value -

P-value from t-score =

p = 2.P(t<-3.75) = 2(0.002812) = 0.005624.

Conclusion -

p < 0.05, hence we reject H₀ at 5% level of significance.

Result :

There is sufficient evidence that average diametre of balls is not equal to 6.7 cms.

c)-

Confidence interval for true mean diametre is -

................... (t is used here, because sample standard deviation is given)

Hence, 95% confidence interval for true mean diametre is (6.4578,6.6422). So, there is 95% chance that the confidence interval (6.4578,6.6422) contains true mean diametre.

d) -

From the test, we conclude that average diametre of balls is not equal to 6.7 cms. & confidence interval for true mean diametre is (6.4578,6.6422). So, confidence interval does not contain diametre of 6.7 cms. Hence, from both we can say that average diametre of balls is not equal to 6.7 cms.