Exam-Style Questions on Significance

A group of students were asked how they get to school each day. The results are shown in the following table

A \(\chi^2\) test was preformed to see if there was a connection between gender and the method of travelling to school.

(a) Write down what the null hypothesis, H_o, might have been for this test.

(b) Find the expected frequency for the females who cycle to school.

(c) Find the \(\chi^2\) statistic.

(d) State whether or not the null hypothesis is accepted at a 5% significance level. Give a reason for your answer.

Worked Solution

One hundred and sixty students are asked how many portions of fruit or vegetables on average they have per day. The data collected is organized in the following table.

A \(\chi ^2\) test is carried out at the 5% level of significance.

(a) Write down the null hypothesis, H_o, for this test.

(b) Write down the number of degrees of freedom for this test.

(c) Find the \(\chi ^2_{calc}\) value for this data.

(d) Find the expected number of females that have more than 4 portions per day (correct to the nearest integer).

(e) Decide whether H_o must be accepted.

Worked Solution

Ucayali, a multinational technology company that focuses on e-commerce, has a distribution centre in Camelot. As part of a study into healthy lifestyles, the number of cups of coffee drunk by 420 staff members each day were recorded. Results are shown in the table.

Cups of coffee per day

The company conducted a \(\chi^2\) test for independence at a \(5\%\) level of significance.

(a) State the null hypothesis.

(b) Calculate the p-value for this test.

(c) State, giving a reason, whether the null hypothesis should be accepted.

Worked Solution

The older students from Glee High School are required to follow a two year IB Mathematics course. Data were gathered from a sample of 242 students regarding their choice of course. The following data were recorded.

A \(\chi ^2\) test was carried out at the 5% significance level to analyse the relationship between gender and choice of mathematics course.

(a) Write down the null hypothesis, H_o, for this test.

(b) Find the expected value of female students on the Studies course.

(c) Write down the number of degrees of freedom.

(d) Use your graphic display calculator to determine the \(\chi ^2\)_calc value.

(e) Determine whether H_o should be accepted. Justify your answer.

One student is chosen at random from the 242 students.

(f) Find the probability that this student is male.

(g) Find the probability that the student chosen at random is on the Standard course.

Two students are chosen at random from the 242 students.

(h) Find the probability that both are on the Studies course.

(i) Find the probability that neither are on the Higher course.

Worked Solution

The staff of a shop kept records of who bought smart phones during the month of February one year. They looked at the numbers of phones bought by gender and the size of the screens. This information is shown in the table below; S represents the size of the screen in centimetres.

The shop manager wants to use this information to predict the probability of selling these sizes of phone screens for the following month.

(a) Use the table to find the probability that a phone will be bought by a female.

(b) Find the probability that a phone with a screen size of 12cm < S ≤ 16cm will be bought.

(c) Find the probability that a phone with a screen size of 12cm < S ≤ 16cm will be bought by a female.

(d) Find the probability that a phone with a screen size greater than 20cm will be bought given that it is bought by a male.

The manager wants to determine whether the screen size is independent of gender so a chi-squared test is performed at the 1% significance level.

(e) Write down the null hypothesis.

(f) Find the expected frequency for females who bought a screen size of 12cm < S ≤ 16cm to the nearest integer.

(g) Write down the number of degrees of freedom.

(h) Write down the \(\chi ^2\) calculated value.

(i) Determine if the null hypothesis should be accepted. Give a reason for your answer.

Worked Solution

The Scrumptious Sweet Company sell a variety pack of colourful, shaped sweets. The sweets are produced such that 60% are square and 40% are circular. It is known that 20% of the square shaped sweets and 40% of the circular sweets are coloured red.

(a) Show this information in a tree diagram.

A sweet is selected at random.

(b) Find the probability that the sweet is red.

(c) Given that the sweet is red, find the probability it is circular.

The Scrumptious Sweet Company also produce variety packs of Rainbow Gums. Their specifications state that the colours in each variety pack should be distributed as follows.

Inspector Lou Spowels opens a pack of 90 sweets and records the frequency of each colour:

To investigate if the sample is consistent with the company's specifications, Mr Spowels conducts a \(\chi^2\) goodness of fit test. The test is carried out at a 5% significance level.

(d) Write down the null hypothesis for this test.

(e) Copy and complete the following table giving the frequencies correct to one decimal place:

(f) Write down the number of degrees of freedom.

(g) Find the p-value for the test.

(h) State the conclusion of the test. Give a reason for your answer.

Worked Solution

The 2nd Rutherford American Scouts joined the 37th Wolverhampton British Scouts for an International Camp. Skipper Jones is interested to see if the mean height of American Scouts, \( \mu_1\), is the same as the mean height of British Scouts, \( \mu_2\). The information is recorded in the following table.

At the 10% level of significance, a t-test was used to compare the means of the two groups. The data is assumed to be normally distributed and the standard deviations are equal between the two groups.

(a) State the null hypothesis.

(b) State the alternative hypothesis.

(c) Calculate the p-value for this test.

(d) State, giving a reason, whether Skipper Jones should accept the null hypothesis.

Worked Solution

A survey was conducted asking people's opinions on whether mouthwash should be used before or after brushing teeth. The results are shown as follows.

A \(\chi ^2\) test at the 1% significance level was conducted.

(a) State \(H_0\), the null hypothesis for the test;

(b) State \(H_1\), the alternative hypothesis for the test.

(c) Write down the number of degrees of freedom.

(d) Find the expected frequency of people between the age 18 to 30 who believe that mouthwash should be used before brushing.

(e) Find the \(\chi ^2\) statistic;

(f) Find the associated p-value for the test.

(g) Determine, giving a reason, whether \(H_0\) should be accepted.

Worked Solution