Calculate Type I error probability

A question is this type if and only if it asks the student to find or calculate the probability of making a Type I error, given a specific test procedure or critical region.

16 questions

CAIE S2 2002 June Q5
5 To test whether a coin is biased or not, it is tossed 10 times. The coin will be considered biased if there are 9 or 10 heads, or 9 or 10 tails.
  1. Show that the probability of making a Type I error in this test is approximately 0.0215 .
  2. Find the probability of making a Type II error in this test when the probability of a head is actually 0.7.
CAIE S2 2023 June Q6
6 When a child completes an online exercise called a Mathlit, they might be awarded a medal. The publishers claim that the probability that a randomly chosen child who completes a Mathlit will be awarded a medal is \(\frac { 1 } { 3 }\). Asha wishes to test this claim. She decides that if she is awarded no medals while completing 10 Mathlits, she will conclude that the true probability is less than \(\frac { 1 } { 3 }\).
  1. Use a binomial distribution to find the probability of a Type I error.
    The true probability of being awarded a medal is denoted by \(p\).
  2. Given that the probability of a Type II error is 0.8926 , find the value of \(p\).
CAIE S2 2015 June Q2
2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.
  1. State suitable null and alternative hypotheses.
  2. Calculate the probability of a Type I error.
  3. State the significance level of the test.
CAIE S2 2017 June Q4
4 It is claimed that 1 in every 4 packets of certain biscuits contains a free gift. Marisa and André both suspect that the true proportion is less than 1 in 4.
  1. Marisa chooses 20 packets at random. She decides that if fewer than 3 contain free gifts, she will conclude that the claim is not justified. Use a binomial distribution to find the probability of a Type I error.
  2. André chooses 25 packets at random. He decides to carry out a significance test at the \(1 \%\) level, using a binomial distribution. Given that only 1 of the 25 packets contains a free gift, carry out the test.
CAIE S2 2013 June Q7
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 \(p\).
CAIE S2 2008 November Q5
5 Every month Susan enters a particular lottery. The lottery company states that the probability, \(p\), of winning a prize is 0.0017 each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her 12 lottery results in a one-year period. She accepts the null hypothesis \(p = 0.0017\) if she has no wins in the year and accepts the alternative hypothesis \(p > 0.0017\) if she wins a prize in at least one of the 12 months.
  1. Find the probability of the test resulting in a Type I error.
  2. If in fact the probability of winning a prize each month is 0.0024 , find the probability of the test resulting in a Type II error.
  3. Use a suitable approximation, with \(p = 0.0024\), to find the probability that in a period of 10 years Susan wins a prize exactly twice.
CAIE S2 2012 November Q3
3 Joshi suspects that a certain die is biased so that the probability of showing a six is less than \(\frac { 1 } { 6 }\). He plans to throw the die 25 times and if it shows a six on fewer than 2 throws, he will conclude that the die is biased in this way.
  1. Find the probability of a Type I error and state the significance level of the test. Joshi now decides to throw the die 100 times. It shows a six on 9 of these throws.
  2. Calculate an approximate \(95 \%\) confidence interval for the probability of showing a six on one throw of this die.
CAIE S2 2013 November Q6
6 At the last election, 70\% of people in Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is \(35 \%\). In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask 6 people from Apoli, chosen at random, and if more than 3 support the president they will accept Luigi's belief.
  1. Calculate the probability of a Type I error.
  2. If Maria's belief is true, calculate the probability of a Type II error.
  3. In fact 2 of the 6 people say that they support the president. State which error, Type I or Type II, might be made. Explain your answer.
OCR S2 2013 January Q9
9 The random variable \(A\) has the distribution \(\mathrm { B } ( 30 , p )\). A test is carried out of the hypotheses \(\mathrm { H } _ { 0 } : p = 0.6\) against \(\mathrm { H } _ { 1 } : p < 0.6\). The critical region is \(A \leqslant 13\).
  1. State the probability that \(\mathrm { H } _ { 0 }\) is rejected when \(p = 0.6\).
  2. Find the probability that a Type II error occurs when \(p = 0.5\).
  3. It is known that on average \(p = 0.5\) on one day in five, and on other days the value of \(p\) is 0.6 . On each day two tests are carried out. If the result of the first test is that \(\mathrm { H } _ { 0 }\) is rejected, the value of \(p\) is adjusted if necessary, to ensure that \(p = 0.6\) for the rest of the day. Otherwise the value of \(p\) remains the same as for the first test. Calculate the probability that the result of the second test is to reject \(\mathrm { H } _ { 0 }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
Edexcel S4 2003 June Q3
3. A train company claims that the probability \(p\) of one of its trains arriving late is \(10 \%\). A regular traveller on the company's trains believes that the probability is greater than \(10 \%\) and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.1\) and \(\mathrm { H } _ { 1 } : p > 0.1\) and accepts the null hypothesis if \(x \leq 2\).
  1. Find the size of the test.
  2. Show that the power function of the test is $$1 - ( 1 - p ) ^ { 10 } \left( 1 + 10 p + 55 p ^ { 2 } \right) .$$
  3. Calculate the power of the test when
    1. \(p = 0.2\),
    2. \(p = 0.6\).
  4. Comment on your results from part (c).
Edexcel S4 2008 June Q6
  1. A drug is claimed to produce a cure to a certain disease in \(35 \%\) of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.
    1. Write down suitable hypotheses to carry out this test.
    2. Find the probability of making a Type I error.
    The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.
    P (cure)0.20.30.40.5
    P (Type II error)0.5880\(r\)0.8565\(s\)
  2. Calculate the value of \(r\) and the value of \(s\).
  3. Calculate the power of the test for \(p = 0.2\) and \(p = 0.4\)
  4. Comment, giving your reasons, on the suitability of this test procedure.
Edexcel S4 2010 June Q3
  1. A manager in a sweet factory believes that the machines are working incorrectly and the proportion \(p\) of underweight bags of sweets is more than \(5 \%\). He decides to test this by randomly selecting a sample of 5 bags and recording the number \(X\) that are underweight. The manager sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\) and rejects the null hypothesis if \(x > 1\).
    1. Find the size of the test.
    2. Show that the power function of the test is
    $$1 - ( 1 - p ) ^ { 4 } ( 1 + 4 p )$$ The manager goes on holiday and his deputy checks the production by randomly selecting a sample of 10 bags of sweets. He rejects the hypothesis that \(p = 0.05\) if more than 2 underweight bags are found in the sample.
  2. Find the probability of a Type I error using the deputy's test. \section*{Question 3 continues on page 12} The table below gives some values, to 2 decimal places, of the power function for the deputy's test.
    \(p\)0.100.150.200.25
    Power0.07\(s\)0.320.47
  3. Find the value of \(s\). The graph of the power function for the manager's test is shown in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0bc6c296-9cbe-498b-89d9-c034b1b246e4-08_1157_1436_847_260} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  4. On the same axes, draw the graph of the power function for the deputy's test.
    1. State the value of \(p\) where these graphs intersect.
    2. Compare the effectiveness of the two tests if \(p\) is greater than this value. The deputy suggests that they should use his sampling method rather than the manager's.
  6. Give a reason why the manager might not agree to this change.
Edexcel S4 2014 June Q5
5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5 The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) and rejects the null hypothesis if \(x < 3\)
  1. Find the size of the test.
  2. Show that the power function of this test is $$( 1 - p ) ^ { 8 } \left( 36 p ^ { 2 } + 8 p + 1 \right)$$ Table 1 gives values, to 2 decimal places, of the power function for the statistician's test. \begin{table}[h] \end{table} Table 1
  3. On the axes below draw the graph of the power function for the statistician's test.
  4. Find the range of values of \(p\) for which the probability of accepting the coin as unbiased, when in fact it is biased, is less than or equal to 0.4
    \includegraphics[max width=\textwidth, alt={}, center]{1d84c9fc-be67-45be-b439-3111c48ff1cb-09_1143_1209_945_402}
SPS SPS FM Statistics 2024 January Q6
6. A firm claims that no more than \(2 \%\) of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than \(2 \%\). In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the \(5 \%\) significance level, using the null hypothesis \(p = 0.02\).
  1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
  2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says
    " 18 out of 600 is \(3 \%\), so there is evidence that the actual proportion of underweight bags is greater than \(2 \%\)." Criticise this statement.
OCR H240/02 2022 June Q12
12 A firm claims that no more than \(2 \%\) of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than \(2 \%\). In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the \(5 \%\) significance level, using the null hypothesis \(p = 0.02\).
  1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
  2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says
    " 18 out of 600 is \(3 \%\), so there is evidence that the actual proportion of underweight bags is greater than \(2 \%\)." Criticise this statement.
AQA Further AS Paper 2 Statistics 2019 June Q2
1 marks
2 A binomial hypothesis test was carried out at the \(5 \%\) level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.6
& \mathrm { H } _ { 1 } : p > 0.6 \end{aligned}$$ A sample of size 30 was used to carry out the test.
Find the probability that a Type I error was made.
Circle your answer.
[0pt] [1 mark]
\(4.4 \%\) 4.8\% 5.0\% 9.4\%