Photo AI
Suzanne is a member of a sports club - AQA - A-Level Maths Pure - Question 17 - 2018 - Paper 3
Question 17
Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches. 17 (a) After buying a new tennis racket Suzanne plays 10 matche... show full transcript
Worked Solution & Example Answer:Suzanne is a member of a sports club - AQA - A-Level Maths Pure - Question 17 - 2018 - Paper 3
Step 1
17 (a) - Investigate whether Suzanne’s new racket has made a difference
Answer
To solve this problem, we need to set up the null and alternative hypotheses.
Step 1: State the Hypotheses
- Null Hypothesis, : The probability of winning matches with the new racket is equal to 0.5.
- Alternative Hypothesis, : The probability of winning matches with the new racket is not equal to 0.5.
Step 2: Identify the Sample Data
- Total matches played: n = 10
- Matches won: x = 7
- Sample proportion: ( p = \frac{x}{n} = \frac{7}{10} = 0.7 )
Step 3: Calculate the Test Statistic We use the binomial distribution to calculate the probabilities: [ P(X \leq 6) \text{ and } P(X \geq 7) = 1 - P(X \leq 6) ] Using binomial tables or software, we find:
- ( P(X \leq 6) \approx 0.8281 ) This gives us: [ P(X \geq 7) \approx 1 - 0.8281 = 0.1719 \approx 0.172 ]
Step 4: Evaluate the Significance Level
- At the 10% significance level, critical regions are for p < 0.05 or p > 0.95. Since 0.172 is greater than 0.05,
- We fail to reject the null hypothesis.
Conclusion: There is not sufficient evidence to conclude that Suzanne’s new racket has made a difference in her win probability.
Step 2
17 (b) - Find the minimum number of matches she must win
Answer
In this case, we need to determine how many matches Suzanne needs to win out of 20 to conclude that the new squash racket has improved her performance.
Step 1: State the Hypotheses
- Null Hypothesis, : Probability of winning matches with the new racket is 0.5.
- Alternative Hypothesis, : Probability of winning matches with the new racket is greater than 0.5.
Step 2: Set Up the Required Probability We express the requirement in terms of the cumulative probability: [ P(X \geq k) > 0.1\ P(X < k) < 0.9 ] for x = number of wins.
Step 3: Calculate the Cumulative Probability Using cumulative probability tables or calculations for the binomial distribution with n = 20:
- To find the minimum number of wins (k), we calculate: [ P(Y \leq y) < 0.1\ \text{where } Y \text{ is the number of matches won.} ]
- The necessary threshold found is k = 14.
Conclusion: Thus, Suzanne must win at least 14 matches to conclude, at the 10% level of significance, that the new racket has improved her performance.