// Numbas version: finer_feedback_settings {"name": "QM Jan 19-20 Final Exam", "metadata": {"description": "

Instructions to candidates:-

a) questions are to be attempted

b) Answer questions in the spaces provided

d) Approved calculators are allowed

e) Marks for each question are shown

f) The total mark for the paper is 100.

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A food manufacturer produces packets of nuts for baking and snacking; the most expensive product they produce is packets of macadamia nuts. From historical data, the weight of macadamias in each packet follows a normal distribution with a population mean (μ) of 146 g and a standard deviation (σ) of 1.6 g.

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What is the probability that a randomly selected packet will weigh less than 142g. Give your answer to 4 dcimal places.

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What is the probability that a randomly selected packet will weigh between 142g to 149g. Give your answer to 4 decimal places.

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40 packets of macadamia nuts were selected from the production line at random. The mean weight for the sample was 146.8g with a sample standard deviation of 2.65g. Which one is the 95% confidence interval for the true mean weight?

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The table shows the amount each film earned in Sales (Box Office Sales) and the Budget (the total amount spent making the film).

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\n Budget (£ million) \n	\n 152 \n	\n 107 \n	\n 180 \n	\n 118 \n	\n 58 \n	\n 62 \n	\n 144 \n	\n 29 \n
\n Sales (£ million) \n	\n 280 \n	\n 294 \n	\n 293 \n	\n 168 \n	\n 92 \n	\n 163 \n	\n 219 \n	\n 66 \n

Calculate the following to 3 decimal places.

Calculate S_xx

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S_yy

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S_xy

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Calculate the Product Moment Correlation Coefficient (r)

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Coefficient of determination r²

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What does the value of the correlation coefficient (r) calculated in part d) tell you about the relationship between the two variables?

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Which one is the correct regression line of Sales on Budget

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Predict the sales for an Action film with a budget of £120 million. Write your answer in millions.

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A manufacturer owns factories in the UK manufacturing furnitures. Assembling king size bed is a long process and is a bottleneck. The senior management team are looking for ways to speed up the process. They decide to train their staffs, which could help to faster the assembly process. 6 workers take part in the training.

The following table gives the average time in minutes it take a worker to assemble a bed before the training and after the training.

Table 2: Average time to assemble a king size bed before training and after training (in minutes)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\n Worker \n	\n A \n	\n B \n	\n C \n	\n D \n	\n E \n	\n F \n
\n Time Before \n	\n 210 \n	\n 180 \n	\n 195 \n	\n 220 \n	\n 231 \n	\n 199 \n
\n Time After \n	\n 193 \n	\n 182 \n	\n 186 \n	\n 186 \n	\n 220 \n	\n 183 \n

Use a paired t-test at a 5% significance level determine if the training has reduced the time taken to assemble a king size bed.

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Which one is the correct null hypothesis?

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H_o: The time taken to assemble a bed is same as before and after the training

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H_o: The time taken to assemble a bed is not same as before

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H_o: The time taken to assemble a bed less than after then training

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Which one is the correct alternative hypothesis?

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H₁: the time taken to assemble a bed has increased.

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H₁: the time taken to assemble a bed has reduced.

", "

H₁: the time taken to assemble a bed has remained same.

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H₁: the time taken to assemble a bed is not same as before.

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Write down the number of degrees of freedom.

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Write down the critical value at 5% significance level.

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Calculate the sample standard deviation to 2 decimal places.

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Calculate the test statistics to 2 decimal places.

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Which one is the correct decision?

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A bakery is considering introducing a new range of baked goods to their product range. Introducing a new product will require capital investment, so the bakery can only afford to introduce one new product in the next year. They are considering three options.

Option A: Introduce a new range of Vegan cakes
Option B: Introduce a new range of low sugar cakes
Option C: Introduce a new range of gluten free cakes

The bakery will carry out a marketing campaign to promote the new product and attract new customers. The profit in the next year will depend on whether the number of customers stays the same, increases slightly or increases significantly.

Table : expected profit (£)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\n \n	\n Same number of customers \n	\n Slight increase in customers \n	\n Significant increase in customers \n
\n Option A (Vegan) \n	\n 3,000 \n	\n 8,000 \n	\n 9,000 \n
\n Option B (Low Sugar) \n	\n 2,000 \n	\n 6,500 \n	\n 15,000 \n
\n Option C (Gluten Free) \n	\n 4,000 \n	\n 5,000 \n	\n 11,500 \n

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Which one is the best decision using the Maximax Rule?

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Which one is the best decision using the Maximin Rule?

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Which one is the best decision based on the Minimax Regret Rule?

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Which one is the best decision using Hurwicz criterion for alpha=0.6?

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An entrepreneur has started a new business selling handmade stationery. In order to attract new customers, the entrepreneur has rented a small stand at an upcoming business event and is intending to sell Calendars and Notebooks at a low promotional price.

The printing of each calendar will cost the entrepreneur £2.00 and the printing of each notebook will cost £3.00. The entrepreneur has £600 to spend on printing costs.

Each calendar will take the entrepreneur 2 hour to make and each notebook takes 1 hour to make. The entrepreneur has 280 hours to make all the items for the stand before the event.

Another business attending the event has agreed to put 30 calendars on their stand, so the entrepreneur must make a minimum of 30 calendars.

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Which set of constraints best represent the informations above if C represents number of Calendars and N represents number of Notebooks?

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2C + 1N ≤ 600
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2C + 1N ≤ 280
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2C + 1N ≤ 600
C ≥ 30", "2C + 3N ≥ 280
2C + 1N ≤ 600
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If the profit on each calendar is £2.00 and the profit on each notebook is £2.50. State which one represents the profit function?

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Profit = 2.5C + 2N

", "

Profit = 2C + 2.5N

", "

Profit = C + 2.5N

", "

Profit = 2C + N

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Using appropriate method, what is the optimal number of Calendar and Notebook should be made to maximise the profit?

C=[[0]], N=[[1]]

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Claculate the maximum profit.

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A local supermarket offers a lunch meal deal, which is very popular with local workers. Each year the supermarket sells 7200 bottles of orange juice as part of the meal deal. The bottles of juice are purchased from a supplier and stored until needed. The cost of placing an order with the supplier is £5.50. The cost of storage is £1.15 per bottle of juice per year.

Calculate the economic order quantity rounded to the nearest whole number.

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The catering company that supplies the bottles of juice offers a discount for larger orders.

Use the supplier price list (shown below) to determine if it is better for the café to use the EOQ or to use a larger quantity and get a discount on the price.

Supplier price schedule

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\n Order size \n	\n Cost per bottle of Orange Juice \n
\n 100 - 500 \n	\n £0.35 \n
\n 501 - 1000 \n	\n £0.32 \n

Calculate the total cost (at EOQ) =[[0]] to 2 decimal places.
Calculate the total cost (at 501)=[[1]] to 2 decimal places.

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Is it cheaper to buy EOQ or larger quantity?

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A pharmaceutical company run a clinical trial for a new flu vaccine. 1200 people volunteered to take part. 625 of the volunteers were given the real flu vaccine. The remaining 575 volunteers received a harmless placebo drug (which in principle would have no positive or negative effect).

The pharmaceutical company believe that the flu vaccine could cause dizziness in some adults.

165 of the volunteers who had the real flu vaccine reported dizziness.

81 of the volunteers who had the placebo drug reported dizziness.

Determine the probability that a volunteer will not suffer dizziness. Give your answer to 3 decimal places.

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Given that a person has dizziness, what is the probability that they have had the real flu vaccine? Give your answer to 3 decimal places.

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The pharmaceutical company find that 1% of people who have the vaccine suffer a more serious side effect (fainting). The company take a random sample of 40 people.

What is the probability that exactly 2 people have the serious side effect in this sample of 40? Give your answer to 3 decimal places.

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What is the probability that at least 2 people have the serious side effect in this sample of 40? Give your answer to 3 decimal places.

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A holiday tour provider gives a questionnaire to a sample of customers returning from a package holiday at a popular tourist destination. 200 customers were asked to rate their holiday accommodation on the following scale; Fantastic, Good, Disappointing. The customer responses for two age groups are shown in the table below (Age group 25-35 and 40-50 years).

Table: Feedback from the holiday questionnaire

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\n \n	\n Age Group \n
\n Accommodation Rating \n	\n 25 - 35 years \n	\n 40 - 50 years \n
\n Fantastic \n	\n 23 \n	\n 31 \n
\n Good \n	\n 36 \n	\n 44 \n
\n Disappointing \n	\n 48 \n	\n 18 \n

Use a Chi-squared test at a 1% significance level to investigate if there is an association between Accommodation Rating and Age Group.

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Which one is the correct null hypothesis?

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H₀: there is no association between Rating and Age

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H₀: there is association between Rating and Age

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H₀: it is impossible to have association between Rating and Age

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H₀: none of the avobe statements are correct

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Which one is the correct alternative hypothesis?

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H₁: there is no association between Rating and Age

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H₁: there is an association between Rating and Age

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H₁: it is impossible to have association between Rating and Age

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H₁: none of the avobe statements are correct

What is the degrees of freedom?

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What is the critical value at 1% significance level?

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Calculate the test statistics to 3 decimal places.

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What is the correct decision?

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