Standard +0.3 This is a standard S2 hypothesis testing question requiring binomial table lookups and understanding of critical regions. While it involves multiple parts, each step follows routine procedures taught in the specification with no novel problem-solving required. The two-tailed test and discrete distribution add minor complexity above the average A-level question.
6. (a) Define the critical region of a test statistic.
A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
(b) Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
(c) Write down the actual significance level of the test.
The value of the observation was found to be 15 .
(d) Comment on this finding in light of your critical region.
(a) The set of values of the test statistic for which the null hypothesis is rejected in a hypothesis test.
B1 B1
(2 marks)
(b) \(X \sim B(30, 0.3)\)
M1
\(P(X \leq 3) = 0.0093\)
(c) Actual significance level \(0.0021 + 0.0064 = 0.0085\) or \(0.85\%\)
B1
(1 mark)
(d) 15 (it) is not in the critical region; not significant; No significant evidence of a change in \(p = 0.3\); accept \(H_0\) (reject \(H_1\)); \(P(x \geq 15) = 0.0169\)
B1ft B1 B1
(2 marks)
Total [10]
**(a)** The set of values of the test statistic for which the null hypothesis is rejected in a hypothesis test. | B1 B1 | (2 marks)
**(b)** $X \sim B(30, 0.3)$ | M1 | $P(X \leq 3) = 0.0093$ | A1 | $P(X \leq 2) = 0.0021$ | A1 | $P(X \geq 16) = 1 - 0.9936 = 0.0064$ | A1 | $P(X \geq 17) = 1 - 0.9979 = 0.0021$ | A1 | Critical region is $(0 \leq) x \leq 2$ or $16 \leq x(\leq 30)$ | A1 A1 | (5 marks)
**(c)** Actual significance level $0.0021 + 0.0064 = 0.0085$ or $0.85\%$ | B1 | (1 mark)
**(d)** 15 (it) is not in the critical region; not significant; No significant evidence of a change in $p = 0.3$; accept $H_0$ (reject $H_1$); $P(x \geq 15) = 0.0169$ | B1ft B1 B1 | (2 marks)
| **Total [10]**
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6. (a) Define the critical region of a test statistic.
A discrete random variable $X$ has a Binomial distribution $\mathrm { B } ( 30 , p )$. A single observation is used to test $\mathrm { H } _ { 0 } : p = 0.3$ against $\mathrm { H } _ { 1 } : p \neq 0.3$\\
(b) Using a $1 \%$ level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005\\
(c) Write down the actual significance level of the test.
The value of the observation was found to be 15 .\\
(d) Comment on this finding in light of your critical region.\\
\hfill \mbox{\textit{Edexcel S2 2010 Q6 [10]}}