5.05d Confidence intervals: using normal distribution

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1. The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was 41.2 minutes and an unbiased estimate of the population variance was 32.6 minutes \({ } ^ { 2 }\). Find a \(95 \%\) confidence interval for the mean time taken to complete the task.
2. The probability that an \(\alpha \%\) confidence interval includes only values that are lower than the population mean is \(\frac { 1 } { 16 }\). Find the value of \(\alpha\).

3 Past experience has shown that the heights of a certain variety of rose bush have been normally distributed with mean 85.0 cm . A new fertiliser is used and it is hoped that this will increase the heights. In order to test whether this is the case, a botanist records the heights, \(x \mathrm {~cm}\), of a large random sample of \(n\) rose bushes and calculates that \(\bar { x } = 85.7\) and \(s = 4.8\), where \(\bar { x }\) is the sample mean and \(s ^ { 2 }\) is an unbiased estimate of the population variance. The botanist then carries out an appropriate hypothesis test.

1. The test statistic, \(z\), has a value of 1.786 correct to 3 decimal places. Calculate the value of \(n\).
2. Using this value of the test statistic, carry out the test at the \(5 \%\) significance level.

1 The weights, in grams, of packets of sugar are distributed with mean \(\mu\) and standard deviation 23. A random sample of 150 packets is taken. The mean weight of this sample is found to be 494 g . Calculate a 98\% confidence interval for \(\mu\).

3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.

7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.

1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , calculate the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 The probability that a certain spinner lands on red on any spin is \(p\). The spinner is spun 140 times and it lands on red 35 times.

1. Find an approximate \(96 \%\) confidence interval for \(p\).
   From three further experiments, Jack finds a 90\% confidence interval, a 95\% confidence interval and a 99\% confidence interval for \(p\).
2. Find the probability that exactly two of these confidence intervals contain the true value of \(p\).

5 A builders' merchant sells stones of different sizes.

1. The masses of size \(A\) stones have standard deviation 6 grams. The mean mass of a random sample of 200 size \(A\) stones is 45 grams. Find a 95\% confidence interval for the population mean mass of size \(A\) stones.
2. The masses of size \(B\) stones have standard deviation 11 grams. Using a random sample of size 200, an \(\alpha \%\) confidence interval for the population mean mass is found to have width 4 grams. Find \(\alpha\).

2 In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha \%\) confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487 , correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer.

7 A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5 . She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. $$n = 50 \quad \Sigma x = 23.0 \quad \Sigma x ^ { 2 } = 13.02$$

1. Carry out a test at the \(5 \%\) significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\).
   Later, a similar test is carried out at the \(5 \%\) significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
2. Given that, in fact, the value of \(\mu\) is 0.4 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The length, in minutes, of mathematics lectures at a certain college has mean \(\mu\) and standard deviation 8.3.

1. The total length of a random sample of 85 lectures was 4590 minutes. Calculate a 95\% confidence interval for \(\mu\).
   The length, in minutes, of history lectures at the college has mean \(m\) and standard deviation \(s\).
2. Using a random sample of 100 history lectures, a 95\% confidence interval for \(m\) was found to have width 2.8 minutes. Find the value of \(s\).

1 The heights of a certain species of deer are known to have standard deviation 0.35 m . A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m .

1. Calculate a 96\% confidence interval for the population mean height.
2. Bubay says that \(96 \%\) of deer of this species are likely to have heights that are within this confidence interval. Explain briefly whether Bubay is correct.

2 The lengths of a random sample of 50 roads in a certain region were measured.Using the results,a \(95 \%\) confidence interval for the mean length,in metres,of all roads in this region was found to be[245,263].

1. Find the mean length of the 50 roads in the sample.
2. Calculate an estimate of the standard deviation of the lengths of roads in this region.
3. It is now given that the lengths of roads in this region are normally distributed.
   State,with a reason,whether this fact would make any difference to your calculation in part(b).

1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .

1. Calculate a 99\% confidence interval for the population mean length.
2. Write down the probability that the whole of a \(99 \%\) confidence interval will lie below the population mean.

4 Packets of cat food are filled by a machine.

1. In a random sample of 10 packets, the weights, in grams, of the packets were as follows. \(\begin{array} { l l l l l l l l l l } 374.6 & 377.4 & 376.1 & 379.2 & 371.2 & 375.0 & 372.4 & 378.6 & 377.1 & 371.5 \end{array}\) Find unbiased estimates of the population mean and variance.
2. In a random sample of 200 packets, 38 were found to be underweight. Calculate a \(96 \%\) confidence interval for the population proportion of underweight packets.

3 A survey of a random sample of \(n\) people found that 61 of them read The Reporter newspaper. A symmetric confidence interval for the true population proportion, \(p\), who read The Reporter is \(0.1993 < p < 0.2887\).

1. Find the mid-point of this confidence interval and use this to find the value of \(n\).
2. Find the confidence level of this confidence interval.

1 Packets of fish food have weights that are distributed with standard deviation 2.3 g . A random sample of 200 packets is taken. The mean weight of this sample is found to be 99.2 g . Calculate a \(99 \%\) confidence interval for the population mean weight.

6 The daily takings, \(\\) x\(, for a shop were noted on 30 randomly chosen days. The takings are summarised by \)\Sigma x = 31500 , \Sigma x ^ { 2 } = 33141816$.

1. Calculate unbiased estimates of the population mean and variance of the shop's daily takings.
2. Calculate a \(98 \%\) confidence interval for the mean daily takings. The mean daily takings for a random sample of \(n\) days is found.
3. Estimate the value of \(n\) for which it is approximately \(95 \%\) certain that the sample mean does not differ from the population mean by more than \(\\) 6$.

2 The weights in grams of oranges grown in a certain area are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). A random sample of 50 of these oranges was taken, and a \(97 \%\) confidence interval for \(\mu\) based on this sample was (222.1, 232.1).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Estimate the sample size that would be required in order for a \(97 \%\) confidence interval for \(\mu\) to have width 8 .

2 A random sample of \(n\) people were questioned about their internet use. 87 of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is \(0.1129 < p < 0.1771\).

1. Write down the mid-point of this confidence interval and hence find the value of \(n\).
2. This interval is an \(\alpha \%\) confidence interval. Find \(\alpha\).

2 In a random sample of 70 bars of Luxcleanse soap, 18 were found to be undersized.

1. Calculate an approximate \(90 \%\) confidence interval for the proportion of all bars of Luxcleanse soap that are undersized.
2. Give a reason why your interval is only approximate.

3 In a sample of 50 students at Batlin college, 18 support the football club Real Madrid.

1. Calculate an approximate \(98 \%\) confidence interval for the proportion of students at Batlin college who support Real Madrid.
2. Give one condition for this to be a reliable result.

7 The weights, \(X\) kilograms, of bags of carrots are normally distributed. The mean of \(X\) is \(\mu\). An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$

1. Carry out the test, at the \(10 \%\) significance level.
2. You may now assume that the population variance of \(X\) is 1.85 . The inspector weighs another random sample of 200 bags and carries out the same test at the \(10 \%\) significance level.
   1. State the meaning of a Type II error in this context.
   2. Given that \(\mu = 2.12\), show that the probability of a Type II error is 0.652 , correct to 3 significant figures.

1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .

1. Calculate a \(99 \%\) confidence interval for the population mean length.
2. Write down the probability that the whole of a \(99 \%\) confidence interval will lie below the population mean.

7 Leila suspects that a particular six-sided die is biased so that the probability, \(p\), that it will show a six is greater than \(\frac { 1 } { 6 }\). She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.

1. State what is meant by a Type I error in this situation and calculate the probability of a Type I error.
2. Assuming that the value of \(p\) is actually \(\frac { 2 } { 3 }\), calculate the probability of a Type II error. Leila now throws the die 80 times and it shows a six on 50 throws.
3. Calculate an approximate \(96 \%\) confidence interval for \(p\).

4 The masses, in grams, of a certain type of plum are normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The masses, \(m\) grams, of a random sample of 150 plums of this type were found and the results are summarised by \(\Sigma m = 9750\) and \(\Sigma m ^ { 2 } = 647500\).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Calculate a 98\% confidence interval for \(\mu\). Two more random samples of plums of this type are taken and a \(98 \%\) confidence interval for \(\mu\) is calculated from each sample.
3. Find the probability that neither of these two intervals contains \(\mu\).