Questions: The numbers of false fire alarms were counted each month at a number of sites. The results are given in the following table. Can you conclude that false alarms are not equally likely to occur in any month? Use the α=0.01 level of significance and the P-value method with the TI- 84 Plus calculator. Month Number of Alarms January 31 February 26 March 38 April 42 May 40 June 28 July 46 August 45 September 47 October 41 November 29 December 46 Part 1 of 4 (a) State the null and alternate hypotheses. H0: p1=p2=p3=…=p10=p11=p12=0.1 H1 : Some or all of the actual probabilities differ from those specified by H0.

The numbers of false fire alarms were counted each month at a number of sites. The results are given in the following table. Can you conclude that false alarms are not equally likely to occur in any month? Use the α=0.01 level of significance and the P-value method with the TI- 84 Plus calculator. Month Number of Alarms January 31 February 26 March 38 April 42 May 40 June 28 July 46 August 45 September 47 October 41 November 29 December 46 Part 1 of 4 (a) State the null and alternate hypotheses. H0: p1=p2=p3=…=p10=p11=p12=0.1 H1 : Some or all of the actual probabilities differ from those specified by H0.
Transcript text: The numbers of false fire alarms were counted each month at a number of sites. The results are given in the following table. Can you conclude that false alarms are not equally likely to occur in any month? Use the $\alpha=0.01$ level of significance and the $P$-value method with the TI- 84 Plus calculator. \begin{tabular}{lc} \hline \multicolumn{1}{c}{ Month } & Number of Alarms \\ \hline January & 31 \\ February & 26 \\ March & 38 \\ April & 42 \\ May & 40 \\ June & 28 \\ July & 46 \\ August & 45 \\ September & 47 \\ October & 41 \\ November & 29 \\ December & 46 \\ \hline \end{tabular} Part 1 of 4 (a) State the null and alternate hypotheses. \[ H_{0}: p_{1}=p_{2}=p_{3}=\ldots=p_{10}=p_{11}=p_{12}=0.1 \] $H_{1}$ : Some or all of the actual probabilities differ from those specified by $H_{0}$.
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Solution

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Solution Steps

Step 1: State the Null and Alternative Hypotheses
  • $H_0$: The event frequencies are equally distributed across all categories ($p_1 = p_2 = \ldots = p_k$).
  • $H_1$: The event frequencies are not equally distributed across all categories (At least one $p_i$ differs).
Step 2: Calculate Expected Frequencies
  • Under $H_0$, each category should have an expected frequency of 38.25.
Step 3: Compute the Test Statistic
  • The Chi-square test statistic is calculated as: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} = 17.261$
Step 4: Determine the P-value
  • The P-value is 0.1.
Step 5: Make a Decision
  • Since the P-value (0.1) is greater than $\alpha$ (0.01), we do not reject $H_0$.

Final Answer: There is not enough evidence to conclude that the event frequencies are not equally distributed across all categories.

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