The lifetime, L, hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours - Edexcel - A-Level Maths Mechanics - Question 5 - 2018 - Paper 2

After using the calculator for 16 hours, Alice has 4 hours left of exams. We first need the likelihood that at least one of the 4 batteries will last more than 20 hours (16 used + 4 remaining).

Using the previously calculated probability of a battery lasting longer than 20 hours:

$Z = \frac{20 - 18}{4} = 0.5$

Finding the corresponding probability for $Z = 0.5$:

$P(Z < 0.5) \approx 0.6915$

The probability of a battery lasting less than 20 hours is:

$P(Z > 0.5) = 1 - P(Z < 0.5) \approx 1 - 0.6915 = 0.3085$

Thus, the probability that all 4 batteries last more than 20 hours is:

$P(all > 20) = (1 - 0.3085)^4 \approx 0.6915^4 \approx 0.2271$

With 2 new batteries added in, the overall probability can be evaluated by focusing on the chances of at least one battery still working:

Given the previous computation, we denote the probability that her calculator does stop working (i.e., all batteries fail) as:

$P(all fail) = 0.3085 ^ 2 + (some mixed cases)$

Then using the complementary probability:

$P(not stop working) = 1 - P(all fail) \approx 0.199$

The hypotheses for the test regarding the battery lifetime are:

• Null Hypothesis ($H_0$): The mean lifetime of batteries is $\mu = 18$ hours.
• Alternative Hypothesis ($H_a$): The mean lifetime of batteries is $\mu > 18$ hours.

Using the sample mean(\bar{x} = 19.2), sample size $n = 20$, standard deviation $\sigma = 4$. We utilize:

$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Substituting in:

$Z = \frac{19.2 - 18}{4 / \sqrt{20}} \approx 1.3416$

This Z-score can be checked against Z-tables to determine if it falls within the critical region for $\alpha = 0.05$. The critical value corresponding to a one-tailed test of 5% significance is approximately 1.645. Since $Z < 1.645$, we fail to reject the null hypothesis, suggesting that Alice's belief does not hold up at the desired level of significance.