A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, Dml, follows a normal distribution with mean 25 ml Given that 15% of bottles contain less than 24.63 ml (a) find, to 2 decimal places, the value of k such that $P(24.63 < D < k) = 0.45$ A random sample of 200 bottles is taken. (b) Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and k ml The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml (c) Test Hannah's belief at the 5% level of significance. You should state your hypotheses clearly.

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A machine puts liquid into bottles of perfume - Edexcel - A-Level Maths Statistics - Question 5 - 2019 - Paper 1

Question 5

A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, Dml, follows a normal distribution with mean 25 ml Given that 15% of bottl... show full transcript

Worked Solution & Example Answer:A machine puts liquid into bottles of perfume - Edexcel - A-Level Maths Statistics - Question 5 - 2019 - Paper 1

Step 1

find, to 2 decimal places, the value of k such that $P(24.63 < D < k) = 0.45$

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Answer

To find the value of k, we first standardize the variable. Given that the mean, (\mu = 25) ml and the corresponding value for 24.63 ml:

Standardize 24.63 ml: $z = \frac{24.63 - 25}{\sigma} = -1.0364$
From the standard normal distribution, we know that 15% of bottles contain less than 24.63 ml. Therefore, we set: $P(Z < -1.0364) = 0.15$
We want to find k such that: $P(24.63 < D < k) = 0.45$ This can be expressed as: $P(D < k) = P(D < 24.63) + 0.45 = 0.15 + 0.45 = 0.60$
Now, we find from the standard normal table: $P(Z < z_k) = 0.60 \Rightarrow z_k = 0.2533$
Reversing the z-score calculation: $k = \mu + z_k \cdot \sigma = 25 + 0.2533 \cdot \sigma$ Using (\sigma = 0.357): $k = 25 + 0.2533 \cdot \sigma \approx 25.09$ Therefore, the value of k is approximately 25.09 ml.

Step 2

Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and k ml

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Answer

Given the sample size of 200 bottles, we define a new variable Y:

Define the distribution: (Y \sim B(200, 0.45)) which will be approximated as (Y \sim N(90, 49.5)) since:
- Mean (\mu = np = 200 \cdot 0.45 = 90)
- Variance (\sigma^2 = np(1-p) = 200 \cdot 0.45 \cdot 0.55 = 49.5)
Probability that fewer than half (100 bottles) contain between 24.63 ml and k ml: (\Rightarrow P(Y < 100) = P\left(Z < \frac{100 - 90}{\sqrt{49.5}}\right))
Calculating the z-score: $z = \frac{100 - 90}{\sqrt{49.5}} \approx 1.4142$
Using the standard normal distribution table:
- Find the probability: $P(Z < 1.4142) \approx 0.9115 \quad (\text{or } 0.912 \text{ when approximated})$

Step 3

Test Hannah's belief at the 5% level of significance. You should state your hypotheses clearly.

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Answer

State the null hypothesis and alternative hypothesis:
- Null Hypothesis: (H_0: \mu \geq 25) (Hannah's belief is not supported)
- Alternative Hypothesis: (H_1: \mu < 25) (Hannah's belief is supported)
Given the sample mean (\bar{x} = 24.94) and sample size (n = 20):
- Standard deviation (s = 0.16)
Calculate the test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{24.94 - 25}{0.16 / \sqrt{20}} \approx -1.614$
Determine the critical value from the t-distribution table for (df = 19) at the 5% significance level (one-tailed test):
- Critical value approximately -1.729.
Conclusion: Since (-1.614 > -1.729), we fail to reject the null hypothesis. There is insufficient evidence to support Hannah's belief.

A machine puts liquid into bottles of perfume - Edexcel - A-Level Maths Statistics - Question 5 - 2019 - Paper 1

Worked Solution & Example Answer:A machine puts liquid into bottles of perfume - Edexcel - A-Level Maths Statistics - Question 5 - 2019 - Paper 1

find, to 2 decimal places, the value of k such that $P(24.63 < D < k) = 0.45$

Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and k ml

Test Hannah's belief at the 5% level of significance. You should state your hypotheses clearly.

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