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Statistics, SPEC1 2023 VCAA 6

Josie travels from home to work in the city. She drives a car to a train station, waits, and then rides on a train to the city. The time, minutes, taken to drive to the station is normally distributed with a mean of 20 minutes and standard deviation of 6 minutes . The waiting time, minutes, for a train is normally distributed with a mean of 8 minutes and standard deviation of minutes . The time, minutes, taken to ride on a train to the city is also normally distributed with a mean of 12 minutes and standard deviation of 5 minutes . The three times are independent of each other.

  1. Find the mean and standard deviation of the total time, in minutes, it takes for Josie to travel from home to the city.   (2 marks)
  1. Josie's waiting time for a train on each work day is independent of her waiting time for a train on any other work day. The probability that, for 12 randomly chosen work days, Josie's average waiting time is between 7 minutes 45 seconds and 8 minutes 30 seconds is equivalent to , where and and are real numbers.
  2. Find the values of and .   (2 marks)
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Statistics, SPEC2-NHT 2019 VCAA 6

A paint company claims that the mean time taken for its paint to dry when motor vehicles are repaired is 3.55 hours, with a standard deviation of 0.66 hours.

Assume that the drying time for the paint follows a normal distribution and that the claimed standard deviation value is accurate.

  1. Let the random variable    represent the mean time taken for the paint to dry for a random sample of 36 motor vehicles.

     

    Write down the mean and standard deviation of  .   (2 marks)

At a car crash repair centre, it was found that the mean time taken for the paint company's paint to dry on randomly selected vehicles was 3.85 hours. The management of this crash repair centre was not happy and believed that the claim regarding the mean time taken for the paint to dry was too low. To test the paint company's claim, a statistical test was carried out.

  1. Write down suitable null and alternative hypotheses and respectively to test whether the mean time taken for the paint to dry is longer than claimed.   (1 mark)

  2. Write down an expression for the    value of the statistical test and evaluate it correct to three decimal places.   (2 marks)

  3. Using a 1% level of significance, state with a reason whether the crash repair centre is justified in believing that the paint company's claim of mean time taken for its paint to dry of 3.55 hours is too low.  (1 mark)

  4. At the 1% level of significance, find the set of sample mean values that would support the conclusion that the mean time taken for the paint to dry exceeded 3.55 hours. Give your answer in hours, correct to three decimal places.  (2 marks)

  5. If the true time taken for the paint to dry is 3.83 hours, find the probability that the paint company's claim is not rejected at the 1% level of significance, assuming the standard deviation for the paint to dry is still 0.66 hours. Give your answer correct to two decimal places.  (1 mark)

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Statistics, SPEC2-NHT 2019 VCAA 19 MC

Bags of peanuts are packed by a machine. The masses of the bags are normally distributed with a standard deviation of three grams.

The minimum size of a sample required to ensure that the manufacturer can be 98% confident that the sample mean is within one gram of the population mean is

  1.  37
  2.  38
  3.  48
  4.  49
  5.  60
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Statistics, SPEC2 2019 VCAA 6

A company produces packets of noodles. It is known from past experience that the mass of a packet of noodles produced by one of the company's machines is normally distributed with a mean of 375 grams and a standard deviation of 15 grams.

To check the operation of the machine after some repairs, the company's quality control employees select two independent random samples of 50 packets and calculate the mean mass of the 50 packets for each random sample.

  1. Assume that the machine is working properly. Find the probability that at least one random sample will have a mean mass between 370 grams and 375 grams. Give your answer correct to three decimal places.   (2 marks)

  2. Assume that the machine is working properly. Find the probability that the means of the two random samples differ by less than 2 grams. Give your answer correct to three decimal places.   (3 marks)

To test whether the machine is working properly after the repairs and is still producing packets with a mean mass of 375 grams, the two random samples are combined and the mean mass of the 100 packets is found to be 372 grams. Assume that the standard deviation of the mass of the packets produced is still 15 grams. A two-tailed test at the 5% level of significance is to be carried out.

  1. Write down suitable hypotheses and for this test.   (1 mark)

  2. Find the    value for the test, correct to three decimal places.   (1 mark)

  3. Does the mean mass of the sample of 100 packets suggest that the machine is working properly at the 5% level of significance for a two-tailed test? Justify your answer.   (1 mark)

  4. What is the smallest value of the mean mass of the sample of 100 packets for to be not rejected? Give your answer correct to one decimal place.   (1 mark)

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Statistics, SPEC2-NHT 2018 VCAA 7

According to medical records, the blood pressure of the general population of males aged 35 to 45 years is normally distributed with a mean of 128 and a standard deviation of 14. Researchers suggested that male teachers had higher blood pressures than the general population of males.

To investigate this, a random sample of 49 male teachers from this age group was obtained and found to have a mean blood pressure of 133.

  1. State two hypotheses and perform a statistical test at the 5% level to determine if male teachers belonging to the 35 to 45 years age group have higher blood pressures than the general population of males. Clearly state your conclusion with a reason.   (3 marks)

  2. Find a 90% confidence interval for the mean blood pressure of all male teachers aged 35 to 45 years using a standard deviation of 14. Give your answers correct to the nearest integer.   (1 mark)

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Statistics, SPEC2 2018 VCAA 6

The heights of mature water buffaloes in northern Australia are known to be normally distributed with a standard deviation of 15 cm. It is claimed that the mean height of the water buffaloes is 150 cm.

To decide whether the claim about the mean height is true, rangers selected a random sample of 50 mature water buffaloes. The mean height of this sample was found to be 145 cm.

A one-tailed statistical test is to be carried out to see if the sample mean height of 145 cm differs significantly from the claimed population mean of 150 cm.

Let denote the mean height of a random sample of 50 mature water buffaloes.

  1. State suitable hypotheses and for the statistical test.   (1 mark)

  2. Find the standard deviation of .   (1 mark)

  3. Write down an expression for the value of the statistical test and evaluate your answer to four decimal places.   (2 marks)

  4. State with a reason whether should be rejected at the 5% level of significance.   (1 mark)

  5. What is the smallest value of the sample mean height that could be observed for to be not rejected? Give your answer in centimetres, correct to two decimal places.   (1 mark)

  6. If the true mean height of all mature water buffaloes in northern Australia is in fact 145 cm, what is the probability that will be accepted at the 5% level of significance? Give your answer correct to two decimal places.   (1 mark)

  7. Using the observed sample mean of 145 cm, find a 99% confidence interval for the mean height of all mature water buffaloes in northern Australia. Express the values in your confidence interval in centimetres, correct to one decimal place.   (1 mark)

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Statistics, SPEC2-NHT 2017 VCAA 20 MC

The mass of suspended matter in the air in a particular locality is normally distributed with a mean of micrograms per cubic metre and a standard deviation of    micrograms per cubic metre. The mean of 100 randomly selected air samples was found to be 40 micrograms per cubic metre.

Based on this, a 90% confidence interval for , correct to two decimal places, is

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