Critical Values/Regions, Population Mean Tests (normal/poisson), Proportion Tests (binomial), Correlation Hypothesis Testing, and Type I/II Errors: notes and practice questions

A renowned artisanal bakery claims that only 5% of its specialty sourdough loaves have minor cosmetic imperfections (e.g., slight cracks, uneven browning). A local restaurant owner, who regularly purchases these loaves, decides to test this claim. For their latest delivery, the owner inspects a batch of 150 loaves and finds 12 loaves with cosmetic imperfections.

(a) Identify the type of sampling used by the restaurant owner.

(b) State the null and alternative hypotheses for this test.

(c) Calculate the p-value for this hypothesis test, assuming cosmetic imperfections occur independently.

(d) The restaurant owner performs the test at the 5% significance level. State the conclusion of the test, giving a reason.

A quality control manager at a manufacturing plant wants to assess the consistency of a new batch of electronic components. He decides to test ten components, ensuring that five are selected from Production Line A and five from Production Line B. The manager instructs the supervisors of each line to provide the required number of components from their current production.

(a) Name the type of sampling that best describes the method used by the quality control manager.

The weights, in grams, of the ten components selected for the test are:

$148, 153, 161, 155, 142, 160, 150, 157, 145, 159$.

(b) For these ten components, find

(i) the mean weight.

(ii) the standard deviation of the weights.

The target weight for these components is $155$ g. The manager is concerned that the components might be consistently underweight. Perform an appropriate test at the $10\%$ significance level to see if the mean weight of the components produced is less than the target weight. It can be assumed that the weights come from a normal population.

State one reason why the test performed in part (c) might not be valid.

Two additional components are tested at a later date. The mean weight for all twelve components is $154.5$ g and the standard deviation is $7.2$ g.

For further analysis, a 'quality score' for the twelve components is obtained by multiplying the weights by $1.5$ and subtracting $50$.

(e) For the twelve components, find

(i) their mean quality score.

(ii) the standard deviation of their quality score.

A bakery owner, "The Daily Loaf", wants to determine if the sales of their signature "Morning Glory Muffin" are consistent throughout the work week. They believe that the number of muffins sold is the same each day.

To test this, they record the number of muffins sold each weekday during a particular week. This data is shown in the table.

Day | Monday | Tuesday | Wednesday | Thursday | Friday

Number of muffins sold | 120 | 135 | 115 | 140 | 150

A goodness of fit test at the 5% significance level is used on this data to determine whether the owner's belief is suitable.

The critical value for the test is 9.49 and the hypotheses are:

$H_0$: The number of Morning Glory Muffins sold is consistent across weekdays.

$H_1$: The number of Morning Glory Muffins sold is not consistent across weekdays.

Find an estimate for how many Morning Glory Muffins the owner expects to sell each day.

Write down the degrees of freedom for this test.

Calculate the $\chi^2$ statistic for this data.

Using the critical value of 9.49, state the conclusion to the test. Give a reason for your answer.

The lifespan of a new type of LED bulb, in hours, can be modelled by a normal distribution with a mean of $12000$ hours and a standard deviation of $800$ hours.

A randomly selected LED bulb is chosen.

(a) Calculate the probability that its lifespan is

(i) less than $11000$ hours.

(ii) between $11500$ hours and $12500$ hours.

(b) $15\%$ of LED bulbs have a lifespan of more than $h$ hours.

Calculate the value of $h$.

A manufacturer wants to determine if a sample of $250$ LED bulbs from a new production batch could have been chosen from a normally distributed population with a mean of $12000$ hours and a standard deviation of $800$ hours.

They perform a $\chi^2$ goodness of fit test at the $5\%$ significance level. They begin by creating the following frequency table:

(c) Calculate, correct to four significant figures, the value of

(i) a.

(ii) b.

The hypotheses for the manufacturer's test are:

$H_0$: The lifespans of the LED bulbs are drawn from a normally distributed population with mean $12000$ hours and standard deviation $800$ hours.

$H_1$: The lifespans of the LED bulbs are not drawn from a normally distributed population with mean $12000$ hours and standard deviation $800$ hours.

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

The critical value for this test is $7.815$.

(e) Perform the $\chi^2$ goodness of fit test and state your conclusion, justifying your reasoning.

A competitor claims that their new 'Brand A' LED bulbs last longer on average than the manufacturer's 'Brand B' LED bulbs.

Random samples of $12$ Brand A bulbs and $10$ Brand B bulbs are chosen, and their lifespans (in hours) are measured:

Lifespans of Brand A bulbs (hours):

Lifespans of Brand B bulbs (hours):

The competitor performs a t-test at the $5\%$ significance level. It is assumed that the populations are normally distributed and have equal variances.

(f) Write down the null and alternative hypotheses for this test.

(g) Perform the t-test and state the conclusion, justifying your reasoning.

A bank manager observes that the number of customers arriving at their main ATM during peak hours follows a Poisson distribution with a mean of 18 customers per hour.

To encourage more foot traffic, the bank launches a new promotional campaign. The manager wants to test if this campaign has increased the number of ATM arrivals. They decide to monitor the ATM for a single 4-hour peak period and use a 5% level of significance for their test.

(a) State the null and alternative hypotheses for the test.

(b) Find the probability that the bank manager will make a Type I error in their test conclusion.

During the 4-hour observation period, the bank manager recorded 85 customer arrivals at the ATM.

(c) State the bank manager's conclusion to the test. Justify your answer.

A company manufactures specialized medical devices. Each device consists of a main circuit board and a protective casing. The weight of the circuit board, $C$, is normally distributed with a mean of $120$ g and a standard deviation of $5$ g. The weight of the protective casing, $P$, is normally distributed with a mean of $30$ g and a standard deviation of $2$ g. The weights of the circuit board and the casing are independent.

Find the probability that a randomly chosen complete device has a total weight of less than $145$ g.

A batch of $10$ such devices is to be packed into a container. The container has a maximum weight capacity of $1510$ g. The weight of each device is independent.

Find the probability that the total weight of the $10$ devices is greater than the capacity of the container.

The company sources a critical microchip component from two different suppliers, Supplier X and Supplier Y. An engineer claims that microchips from Supplier Y have a lower response time than those from Supplier X. To test this claim, a random sample is taken from each supplier.

The eight microchips in the sample from Supplier X have response times, in milliseconds (ms), of:

$19.65, 19.65, 19.79, 20.75, 20.97, 21.15, 22.28, 22.37$.

Find the

(i) mean response time for the sample from Supplier X.

Find the

(ii) unbiased estimate of the population variance for the sample from Supplier X.

The seven microchips in the sample from Supplier Y have a mean response time of $19.5$ ms and an unbiased estimate of the population standard deviation ($s_{n-1}$) of $1.05$ ms.

Perform a suitable test, at the $5\%$ significance level, to test the engineer's claim that microchips from Supplier Y have a lower response time than those from Supplier X. You may assume the response times of microchips from each supplier are normally distributed with equal population variance.

A materials scientist is developing a new synthetic fiber and claims its average tensile strength is 120 N. To test this claim, she takes a random sample of 25 fiber segments.

Given that the tensile strength of an individual fiber segment is $x$ newtons, the scientist found that, from her 25 samples, $\sum x = 2968$ and $\sum x^2 = 352480$.

(a) Find an unbiased estimate for the mean tensile strength ($\mu$) of the fiber.

(b) Use the formula $s_{n-1}^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}$ to determine an unbiased estimate for the variance of the tensile strength of the fiber.

(c) Find a 95% confidence interval for $\mu$. You may assume that all conditions for a confidence interval have been met.

(d) Suggest, with justification, a valid conclusion that the materials scientist could make regarding her claim.

(a) The manager of "The Daily Grind" coffee shop suggested that the number of customers arriving at the shop during a 5-minute interval can be modelled by a Poisson distribution.

Suggest two observations that the manager may have made that led him to suggest this model.

(b) Now assume that the model is valid and that the mean number of customers arriving at the shop during a 5-minute interval is $1.5$.

The manager observes customer arrivals during a 15-minute interval.

Calculate the probability that exactly 6 customers arrive during this 15-minute interval.

(c) Using the same model as in part (b), find the probability that fewer than 4 customers arrive during a 15-minute interval.

(d) Find the probability that in four consecutive 5-minute intervals, at least one customer arrives in each interval.

(e) Following a new marketing campaign, the manager wished to determine whether the mean number of customers arriving during a 5-minute interval had increased.

State the hypotheses for the test.

(f) Find the critical region for the test at the 5% significance level.

(g) Given that the mean number of customers per 5-minute interval has actually risen to $2.5$, find the probability that the manager makes a Type II error.

The number of incoming calls received by a tech support hotline in a particular 30-minute period is historically known to follow a Poisson distribution with a mean of 48 calls per hour.

A new automated system is implemented, and it is claimed that this system has decreased the number of calls requiring human intervention.

To test this claim, the number of calls, Y, requiring human intervention in a 30-minute period on a particular day will be recorded. The test will have the following hypotheses:

$H_0$: the mean number of calls requiring human intervention has not changed,

$H_1$: the mean number of calls requiring human intervention has decreased.

The alternative hypothesis will be accepted if $Y \leq 17$.

Assuming the null hypothesis to be true, state the distribution of Y.

Find the probability of a Type I error.

Find the probability of a Type II error, if the number of calls now follows a Poisson distribution with a mean of 35 calls per hour.

Ms. Anya Sharma, a school principal, wants to investigate if the number of hours students spend studying affects their exam scores. This question asks you to review Ms. Sharma's methods and conclusions.

Ms. Sharma obtained a list of students from her school. She contacted them and asked them to fill in an anonymous questionnaire. Participants were asked to state their weekly study hours and their most recent exam score (out of 100). Of the 250 students on the list, 11 replied.

Ms. Sharma's results are summarized in the following table:

Describe one way in which Ms. Sharma could improve the reliability of her investigation.

Describe one criticism that can be made about the validity of Ms. Sharma's investigation.

Ms. Sharma classifies Student 11 as an outlier and removes their data from the analysis. Suggest one possible justification for her decision to remove it.

For the remaining ten student responses in the table, Ms. Sharma calculates the mean exam score to be $66$. Calculate the mean weekly study hours for these remaining responses.

Determine the value of $r$, Pearson's product-moment correlation coefficient, for these remaining responses.

Ms. Sharma decides to carry out a hypothesis test on the correlation coefficient to investigate whether increased weekly study hours are associated with higher exam scores. State why the hypothesis test should be one-tailed.

State the null and alternative hypotheses for this test.

The critical value for this test, at the 5% significance level, is 0.549. Ms. Sharma assumes that the population is bivariate normal. Determine whether there is significant evidence of a positive correlation between weekly study hours and exam scores. Justify your answer.

Ms. Sharma wants to create a model to predict how changing weekly study hours might affect exam scores. To do this, she assumes that weekly study hours, $X$, is the independent variable and the exam score, $Y$, is the dependent variable.

She first considers a linear model of the form $Y = aX + b$. Use Ms. Sharma's data to find the value of $a$ and of $b$.

Interpret, referring to study hours and exam scores, what the value of $a$ represents.

Ms. Sharma then considers a quadratic model of the form $Y = cX^2 + dX + e$. Find the value of $c$, of $d$ and of $e$.

Find the coefficient of determination for each of the two models she considers.

Hence compare the two models.

Ms. Sharma decides to use the coefficient of determination to choose between these two models. Comment on the validity of her decision.

After presenting the results of her investigation, a colleague questions whether Ms. Sharma's sample is representative of all students in the school. A report states that the mean weekly study hours for all students in the school is $9$ hours. Ms. Sharma decides to carry out a test to determine whether her sample could realistically be taken from a population with a mean of $9$ hours. State the name of the test which Ms. Sharma should use.

State the null and alternative hypotheses for this test.

Perform the test, using a 5% significance level, and state your conclusion in context.

A factory produces two types of electronic components: standard (S) and premium (P). The weight of these components is a critical characteristic for quality control.

The weights of standard components are known to be normally distributed with a mean of 150 grams and a standard deviation of 5 grams.

The weights of premium components are known to be normally distributed with a mean of 165 grams and a standard deviation of 8 grams.

A quality control machine classifies a component as 'premium' if its weight is found to be above 158 grams; otherwise, it is classified as 'standard'.

The factory's quality control manager uses the null hypothesis that, in the absence of other information, a component is standard.

Calculate the probability of making a Type I error when classifying a component.

Calculate the probability of making a Type II error when classifying a component.

It is known that 80% of the components produced are standard, and 20% are premium.

Calculate the overall probability that a randomly selected component is misclassified by the machine.

(a) Mr. Lee, the owner of "Sweet Delights" bakery, recorded the number of Mooncakes sold each day for a sample of $30$ days. The results are shown in the table below.

(a.i) Find the mean and variance for this sample data.

(a.ii) Hence, state why Mr. Lee might believe that the daily sales of Mooncakes follow a Poisson distribution.

(b) State one assumption that Mr. Lee needs to make about the sales of Mooncakes to support his belief that it follows a Poisson distribution.

(c) Mr. Lee knows from his historic sales records that the bakery sells an average of $3.9$ Mooncakes each day. The following table shows the expected frequency of Mooncakes sold each day during a $100$-day period, assuming a Poisson distribution with mean $3.9$.

Find the value of a, of b, and of c. Give your answers to 3 decimal places.

(d) Mr. Lee decides to carry out a $\chi^2$ goodness of fit test at the $5\%$ significance level to see whether the daily sales of Mooncakes follow a Poisson distribution with mean $3.9$. He collects observed frequencies for $100$ days, which are given in the table below.

(d.i) Write down the number of degrees of freedom for his test.

(d.ii) Perform the $\chi^2$ goodness of fit test and state, with reason, a conclusion.

(e) Mr. Lee claims that a new social media advertising campaign, costing $250$ THB per day, will increase the number of Mooncakes sold. However, his business partner, Ms. Chen, claims that the advertising will not increase the bakery's overall profit.

Ms. Chen agrees to run the campaign for the next $40$ days. During that time, Mr. Lee records that the bakery sells a total of $180$ Mooncakes, with a profit of $45$ THB on each Mooncake sold.

Mr. Lee wants to carry out an appropriate hypothesis test to determine whether the number of Mooncakes sold during the $40$ days increased when compared with the historic sales records (mean $3.9$ Mooncakes per day).

By finding a critical value, perform this test at a $5\%$ significance level.

(f) Hence state the probability of a Type I error for this test.

(g) By considering the claims of both Mr. Lee and Ms. Chen, explain whether the advertising campaign was beneficial to the bakery.

A manufacturing company produces electronic components. Historically, the defect rate for a specific component has been 15%. A new production line is implemented, and the quality control manager wants to test if the defect rate has decreased. She assumes that the defect status of each component is independent of others.

(a) Write down suitable hypotheses for this test.

(b) The quality control manager decides to take a random sample of 120 components. She will reject the null hypothesis if fewer than 12 components are found to be defective.

Find the probability that she makes a Type I error.

(c) In fact, the new production line successfully reduced the defect rate to 10%.

Find the probability that she makes a Type II error.

(a) TechInnovate is considering collecting more data for their analysis.

(i) State one advantage of increasing the sample size.

(ii) State one disadvantage of increasing the sample size.

(b) The production manager at Plant Alpha recorded the time, in minutes, taken to produce a batch of electronic components for 10 randomly selected batches:

$18.2, 19.5, 17.8, 20.1, 18.5, 19.0, 17.5, 20.5, 18.8, 19.3$

Find the value of $s_{n-1}$ for this sample from Plant Alpha.

(c) A manager claims that Plant Alpha's production times are more consistent than Plant Beta's. Given that the sample standard deviation ($s_{n-1}$) for Plant Beta's production times is $1.05$ minutes, make one criticism of this claim.

(d) TechInnovate wants to compare the mean production times of Plant Alpha and Plant Beta using a pooled t-test.

(i) State the condition regarding population variances required to use a pooled t-test.

(ii) Given that for Plant Beta, a sample of $12$ batches yielded a mean production time of $\bar{x}_B = 19.3$ minutes and a sample standard deviation of $s_B = 1.05$ minutes, state whether TechInnovate should use a pooled t-test in this case. Justify your answer.

(e) TechInnovate believes Plant Alpha has a lower mean production time than Plant Beta.

(i) State appropriate null and alternative hypotheses for the pooled t-test.

(ii) Find the p-value.

(iii) Given that the test is carried out at the 5% significance level, state the appropriate conclusion in context. Justify your answer.

(f) The company also investigates the relationship between operator experience (in years) and the number of defective items produced per day. A sample of 8 operators yielded the following data:

(i) Assuming all requirements are met, perform a test at the 5% significance level to determine if there is a linear correlation between operator experience and the number of defective items. State the hypotheses and justify your conclusion.

(ii) If the requirements for this test are not met, state an alternative test that could be used.

(g) For the data in (f.i), the equation of the least squares regression line of defective items ($D$) on operator experience ($E$) is $D = -1.5E + 18.25$. Give, in context, an interpretation of the gradient $-1.5$ in this model.

(h) TechInnovate uses a baseline model to predict the number of defective items ($D_{pred}$) for a batch based on its size ($S$): $D_{pred} = 0.5S + 10$. The "Quality Deviation" ($Q$) for a batch is defined as $Q = D_{pred} - D_{actual}$. A positive Quality Deviation indicates better-than-expected quality.

(i) Show that for a batch of $150$ components from Plant Beta that produced $80$ defective items, the Quality Deviation is $5.0$.

(ii) To compare quality control, samples of Quality Deviation scores were collected:

• Plant Alpha: $n_{QA} = 15$, $\bar{x}_{QA} = 4.5$, $s_{QA} = 1.2$

• Plant Beta: $n_{QB} = 18$, $\bar{x}_{QB} = 3.8$, $s_{QB} = 1.1$

Assuming that the appropriate requirements are met, use a pooled t-test at a 5% significance level to determine if the mean Quality Deviation is higher in Plant Alpha than in Plant Beta. Write down your null and alternative hypotheses and justify your conclusion.

(i) Using the results from parts (e) and (h.ii), explain how each plant could claim they are performing better than the other plant.

Ms. Chen, a small business owner, wants to investigate if there is a monotonic relationship between the monthly advertising budget and the number of units of a new product sold. She collects data for eight months, as shown in Table 1.

Table 1: Monthly Advertising Budget and Units Sold

Ms. Chen decides to calculate the Spearman's rank correlation coefficient. Complete the table of ranks shown in Table 2.

Table 2: Ranks for Advertising Budget and Units Sold

Calculate the value of $r_s$, Spearman's rank correlation coefficient.

Ms. Chen believes that a higher advertising budget leads to more units sold. She carries out a hypothesis test using a 10% significance level with the following null hypothesis:

$H_0$: In the population, there is no monotonic relationship between the monthly advertising budget and the number of units sold.

Write down Ms. Chen's alternative hypothesis.

The critical value of $r_c$ for this test is 0.643.

State the conclusion of the hypothesis test, giving a reason.

The battery life, in hours, of a particular smartphone model, $L$, can be modelled by a normal distribution with a mean of 24 hours and a standard deviation of 2 hours.

(a) Find the probability that a randomly selected smartphone has a battery life greater than 27 hours.

Two smartphones are selected at random and independently of each other.

(b) (i) Find the probability that both smartphones have a battery life greater than 27 hours.

(b) (ii) Find the probability that their total battery life is greater than 52 hours.

A software update is released which is claimed to improve battery life. The manufacturer decides to take a random sample of 20 smartphones to test this claim at the 1% significance level, assuming the standard deviation of the battery life has not changed.

(c) Write down the null and alternative hypotheses for the test.

(d) Find the critical region for this test.

Unknown to the manufacturer, the software update has resulted in all smartphones having a 5% longer battery life than the original model.

(e) Find the mean and standard deviation of the battery life for smartphones with the update.

(f) Find the probability of a Type II error in the manufacturer’s test.

A botanist is investigating the effectiveness of two different soil compositions, Soil A and Soil B, on the growth of a particular plant species. They hypothesize that Soil A will lead to a greater mean plant height after four weeks compared to Soil B. The botanist grows a random sample of plants in each soil type and measures their heights (in cm) after four weeks. The results are shown in the table below.

The botanist performs a one-tailed t-test at a 5% level of significance. It is assumed that the plant heights are normally distributed and the samples have equal variances.

State the null and alternative hypotheses.

Calculate the p-value for this test.

State the conclusion of the test in the context of the question. Justify your answer.

The battery life, in hours, of eight Brand A batteries was recorded as $45.2, 47.8, 48.5, 49.1, 50.3, 51.7, 52.4, 54.0$.

The battery life, in hours, of $10$ Brand B batteries was recorded as $43.5, 44.9, 46.1, 47.2, 48.0, 48.8, 49.5, 50.1, 51.0, 52.3$.

(a) Find the sample mean battery life for

(i) Brand A batteries.

(ii) Brand B batteries.

You can assume that both sets of values have a common unknown variance.

(b) Carry out a test at the $10\%$ significance level to determine if the population mean battery life of Brand A is longer than that of Brand B. In your answer, you should include the test used (with a reason), the test hypotheses, the $p$-value and the conclusion of the test (with a reason).

(c) State if your conclusion would have been any different if working at the $5\%$ significance level.