Exercises: Confidence intervals

A social media analyst wants to study the daily time that Irish teenagers aged between 15 and 18 spend on TikTok. To do so, they randomly select 400 users in this age group across the country and record their connection times.

Clearly identify the following elements of the statistical study:

1. a) The population under study.

1. b) The sample selected.

1. c) The sample size.

1. d) The statistic that would be calculated to summarise the information from the sample.

It is known that the waiting time in the queue of a popular theme park ride follows a normal distribution with a mean of $\mu = 45$ minutes and a standard deviation of $\sigma = 12$ minutes.

If we choose a random sample of $n = 36$ visitors who have been on the ride:

1. What distribution does the sample mean $\bar{X}$ follow? State its mean and standard deviation (standard error).

2. Calculate the probability that the mean waiting time of this sample is greater than $48$ minutes.

In statistical inference, the critical value $z_{\alpha/2}$ is essential for determining the width of a confidence interval. It is the value of the standard normal distribution $N(0, 1)$ that leaves an area $\alpha/2$ to its right, where $1-\alpha$ is the confidence level (in the figure, the central area is $1-\alpha$ and each tail is $\alpha/2$).

Using the standard normal distribution table $N(0, 1)$, find the critical values $z_{\alpha/2}$ corresponding to the following confidence levels:

1. a) A confidence level of $90\,\%$.

1. b) A confidence level of $95\,\%$.

1. c) A confidence level of $99\,\%$.

A courier company is studying the weight of packages it delivers in a city. The population is assumed to follow a normal distribution with a standard deviation of $\sigma = 0.8$ kg. A random sample of $n = 64$ packages is taken, obtaining a sample mean of $\bar{x} = 4.2$ kg.

Calculate a confidence interval for the mean weight of all packages delivered by the company, using a confidence level of $95\%$.

After conducting a study on the monthly expenditure on streaming platforms among young people in a neighbourhood, it was found that the confidence interval for the population mean, at a given confidence level, is $(\text{€}\ 14,20, \text{€}\ 17.80)$.

From this interval, determine:

1. The value of the sample mean $\bar{x}$ used for the calculation.

2. The maximum allowable error $E$ made in the estimation.

In a secondary school, it is suspected that $15\%$ of the students are left-handed. A random sample of $n = 100$ students is selected to check this.

1. What distribution does the sample proportion $\hat{p}$ of left-handed students follow? State its mean and standard deviation.

2. Calculate the probability that the proportion of left-handed students in the selected sample is less than $18\%$.

We wish to estimate the proportion of households in a city that separate organic waste for recycling. To do so, a random sample of $n = 400$ households is surveyed, of which $160$ state that they separate this waste.

Determine the confidence interval for the proportion of households that recycle organic waste across the whole city, using a confidence level of $90\%$.

A LED light bulb factory wishes to estimate the mean lifespan of its products. It is known from previous studies that the standard deviation of the lifespan is $\sigma = 500$ hours.

If the quality control team wants to make an estimation with a confidence level of $95\%$, what minimum sample size $n$ should they choose so that the maximum error incurred is less than $100$ hours?

Hugo is carrying out a project for his Economics class on the monthly public transport expenditure of students on his campus. He knows that the variable follows a normal distribution with standard deviation $\sigma = \text{€}\ 15$. Based on a random sample of $100$ students and setting a $95\%$ confidence level ($z_{\alpha/2} = 1.96$), he calculates the margin of error and obtains $E = 1.96 \times \frac{15}{\sqrt{100}} = \text{€}\ 2.94$. Next, Hugo states in his conclusions: “To halve the margin of error ($\text{€}\ 1.47$) while maintaining the same $95\%$ confidence level, it will be sufficient to double the effort and survey $200$ students.” a) Identify the error in Hugo's statement and reasonably explain why doubling the sample size does not divide the error by two. b) Correct the calculation by determining how many students he must actually survey so that the margin of error is $\text{€}\ 1.47$.

In a study on the daily time (in minutes) that 6th Year students spend reading in digital format, it is assumed that the variable follows a normal distribution with a known population standard deviation $\sigma = 18\text{ minutes}$. An official report only publishes the confidence interval obtained for the population mean at a $90\%$ confidence level ($z_{\alpha/2} = 1.645$), which is $(42.04;\ 47.96)$, but it does not specify the sample size or the observed sample mean. a) Reconstruct the sample mean $\bar{x}$ and the margin of error $E$, justifying each step algebraically using the symmetry properties of the interval. b) State and solve the equation that allows you to deduce the sample size $n$ used by the researchers.

For a study on digital reading habits among young people, a confidence interval is to be constructed. Calculate, showing your working, the critical values $z_{\alpha/2}$ corresponding to the following confidence levels:

1. a) $94 \%$
2. b) $98 \%$

Illustrate graphically the position of the critical value in a standard normal distribution for the $94 \%$ case.

An esports team wants to know the mean daily training time of its players. It is known that playing time follows a normal distribution with a standard deviation of $45$ minutes. A random sample of $36$ players from the national league was taken, yielding a mean of $210$ minutes daily.

1. a) Determine the $95 \%$ confidence interval for the mean daily training time of all players in the league.
2. b) What is the maximum margin of error made in this estimate?

A technology company is testing the battery life of a new smartphone model. It is assumed that the duration follows a normal distribution with a standard deviation of $2.4$ hours.

What should be the minimum sample size so that the maximum margin of error made when estimating the population mean is less than $30$ minutes, with a confidence level of $99 \%$?

In a survey carried out on $400$ students at a secondary school about the use of streaming platforms, $260$ said that they prefer watching TV series rather than films.

1. a) Find the point estimate of the proportion of students who prefer series.

1. b) Obtain a $97\,\%$ confidence interval for the true proportion of students who prefer series at that school.

The price of tickets for a summer music festival follows a normal distribution with a standard deviation of $12$ euros. A random sample of prices was taken, obtaining the following confidence interval for the mean: $(72.15; 77.85)$.

1. a) What was the mean price of the selected sample?
2. b) If the confidence level used for this interval was $95 \%$, what was the size of the sample analysed?

An estimate is to be made of the proportion of citizens who would be willing to use public transport if it were free. In a prior study the expected proportion is unknown, so maximum variance ($p = q = 0.5$) is assumed.

Determine the minimum sample size required to estimate this proportion with a maximum margin of error of $4 \%$ and a confidence level of $95.44 \%$ (which corresponds to a critical value of $z_{\alpha/2} = 2$).

Two secondary schools, A and B, measure the mean time that their 2nd year Baccalaureate students spend studying mathematics per week. In both schools, it is known that the population standard deviation is $1.5$ hours.

A sample of $100$ students is taken in each school. In school A, the confidence interval obtained for the mean has a width of $0.588$ hours. In school B, the confidence interval obtained for the mean is $(6.2; 7.2)$.

1. a) Determine the confidence level used in school A.
2. b) Determine the confidence level used in school B.
3. c) Which of the two schools has made a more precise estimate? Justify your answer.

An "eco-friendly" clothing brand claims that $30\,\%$ of its customers are under $20$ years old. To check this, a random sample of $500$ customers is taken and it is observed that $135$ of them are under $20$ years old.

1. a) Construct a $99\,\%$ confidence interval for the proportion of customers under $20$ years old based on the sample data.

1. b) In view of the result, can we consider the brand's claim to be consistent with the data obtained at a $99\,\%$ confidence level? Explain why.

The Road Safety Authority (RSA) wants to estimate the proportion of drivers in a county who regularly use a mobile phone while driving without a hands-free device. In a preliminary pilot survey of $200$ drivers, $32$ stated that they committed this infraction. The final study is to be carried out with a $95\%$ confidence level ($z_{\alpha/2} = 1.96$) and a maximum margin of error of $2\%$ ($E = 0.02$). a) Before using the exact formula, make a reasoned qualitative estimate: knowing that the variance of a proportion is $\hat{p}(1-\hat{p})$, will more or fewer than $1000$ drivers be needed to achieve that precision? Justify your intuition by comparing the variability of this case with the worst-case scenario ($\hat{p} = 0.5$). b) Calculate the minimum sample size $n$ required using the proportion from the pilot sample. c) Check whether the sample size obtained satisfies the validity conditions for the Wald interval for proportions ($n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$), and reflect on whether the final result confirms your estimate from part a).

From the microdata of the Central Statistics Office (CSO) on housing, an analyst models the rental price per square metre in an Irish city such as Limerick. It is assumed that the price per square metre follows a normal distribution with a known population standard deviation $\sigma = 3.50\text{ \text{€}/m}^2$. From a random sample of $49$ dwellings, a sample mean of $\bar{x} = 12.80\text{ \text{€}/m}^2$ is obtained. a) Model the situation by constructing the $95\%$ confidence interval ($z_{\alpha/2} = 1.96$) for the population mean rental price per square metre. b) A representative of a neighbourhood association claims: "There is a $95\%$ probability that the true average rental price in the city lies within that numerical interval you calculated." Explain why this interpretation is conceptually flawed from the perspective of frequentist inference, and write down the correct interpretation. c) If the Department of Housing, Local Government and Heritage claimed that the average price in that area is $11.50\text{ \text{€}/m}^2$, justify whether this value is plausible in light of the interval obtained.

A well-known content creator is analysing how long his followers stay connected during his live e-sports streams. The connection time is known to follow a normal distribution with a standard deviation of $12$ minutes. After analysing a random sample of $144$ users, a confidence interval for the population mean of $(32.35; 36.25)$ has been obtained.

1. a) Determine the value of the sample mean obtained and the confidence level used to construct this interval.

1. b) If the streamer wants to reduce the width of this interval to one quarter while keeping the same confidence level, what should the minimum sample size be?

1. c) Explain, without carrying out any additional calculations, what would happen to the precision of the estimate if we increased the sample size but also wanted to work with a confidence level higher than the one calculated in part a).

A mobile phone company wants to estimate the proportion of teenagers who use "dark mode" on their devices at night. In an initial sample of $100$ users, $60\%$ confirmed using it.

1. a) Calculate the $98\%$ confidence interval for the population proportion of users of this mode.

1. b) Suppose that no prior information about the proportion is available (maximum uncertainty scenario). Determine the sample size needed for the maximum margin of error to be $0.02$ with a confidence level of $95\%$.

1. c) If the proportion obtained in the initial sample ($0.6$) is used to calculate the sample size under the same conditions as in part b) (margin of error $0.02$ and confidence level $95\%$), how many fewer users would need to be surveyed compared to the maximum uncertainty scenario?

The reaction time of pilots using a state-of-the-art flight simulator follows a normal distribution with unknown mean and standard deviation of $0.05$ seconds. On taking a sample of $64$ pilots, a mean of $0.42$ seconds is observed.

1. a) Obtain the confidence interval for the population mean at a confidence level of $97\%$.

1. b) The simulator manufacturer claims that the mean reaction time is $0.40$ seconds. Is this claim compatible with the interval obtained in the previous part? Justify your answer based on the relationship between confidence intervals and population parameters.

1. c) If it is decided to double the sample size and reduce the confidence level to $90\%$, how will this affect the width of the interval? Calculate the factor by which the width is reduced or increased.

A study on sustainable habits reveals that $15\%$ of young people in a city travel exclusively by electric scooter. A random sample of $n$ young people is selected for follow-up.

1. a) If the confidence interval obtained for the population proportion at a confidence level of $95\%$ has a width of $0.06$, what was the sample size used? (Use the proportion of $15\%$ as the estimate).

1. b) If we wanted the margin of error to be only $0.01$ while maintaining $95\%$ confidence, but suspect that the true proportion might be closer to $50\%$, what would be the new required sample size?

The weight of the backpacks of 2nd year Baccalaureate students follows a normal distribution with a standard deviation of $1.2$ kg. A random sample of $100$ students was taken and a confidence interval for the mean weight was constructed. The upper bound of this interval is $6.235$ kg and its width is $0.47$ kg.

1. a) What is the sample mean weight of the backpacks in this study?

1. b) Determine the confidence level with which the interval was constructed.

1. c) If the maximum margin of error is required to be $0.1$ kg with a confidence level of $99.73\%$ (corresponding to $3\sigma$), how many additional students should be included in the original sample?

A survey was conducted among $400$ users of a music streaming platform to find out what percentage prefers urban music. The confidence interval obtained for the proportion of users is $(0.312; 0.388)$.

1. a) Determine the value of the sample proportion and the confidence level used in the study.

1. b) If it is decided to conduct a new study with a sample of $1600$ users and the sample proportion remains the same, what will be the width of the new interval if working with the same confidence level?

1. c) Justify mathematically how the width of the interval changes if we quadruple the sample size and double the critical value $z_{\alpha/2}$.

A technology centre is researching the battery life of a new smartphone model under heavy use. It is assumed that the duration follows a normal distribution with $\sigma = 0.8$ hours. It is desired to estimate the mean duration with $96\%$ confidence.

1. a) If the confidence interval is required to have a maximum length of $20$ minutes, calculate the minimum required sample size.

1. b) After conducting the experiment with the sample size obtained in the previous part, the sample mean is observed to be $9.5$ hours. Write the resulting confidence interval in hours.

1. c) If, due to budget constraints, only $50$ devices could be analysed, what would be the maximum margin of error incurred if we wished to maintain $96\%$ confidence? Express the result in minutes.

In a university entrance examination, the marks obtained by students in Applied Mathematics for Social Sciences II follow a normal distribution with a standard deviation of $1.5$ points. Two independent samples are selected from different secondary schools, both of size $n$.

• For Sample A, a confidence level of $90\%$ is used, yielding a margin of error of $0.25$ points.

• For Sample B, a confidence level of $99\%$ is required.

1. a) Calculate the sample size $n$ used in Sample A.

1. b) Using the value of $n$ found in the previous part, determine the maximum admissible margin of error for Sample B.

1. c) If it were decided to combine both samples into a single sample of size $2n$, calculate the new margin of error for a confidence level of $95\%$. Is this margin of error smaller than that of Sample A? Reason whether precision always improves when increasing the sample size regardless of changes in the confidence level.

A sociological consultancy firm must design a survey to estimate the proportion of young people aged 18 to 25 suffering from digital gambling addiction in a county. Based on previous nationwide polls, the true proportion is suspected to be small, around $p \approx 0.02$ ($2\%$). The technical team is debating between two methodological options: Proposal A, take a sample of $n_A = 150$ young people with a $99\%$ confidence level ($z_{\alpha/2} = 2.575$); Proposal B, take a sample of $n_B = 600$ young people with a $90\%$ confidence level ($z_{\alpha/2} = 1.645$). a) Evaluate the technical conditions for the normal approximation of the Wald interval ($n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$) in both proposals under the assumption that the sample reflects the expected value. What numerically absurd result occurs at the lower bound of the interval if the formula is applied mechanically in Proposal A? b) Calculate the theoretical margin of error for each proposal. c) Give a reasoned justification of which of the two methods the consultancy firm should choose if it prioritises the inferential validity and practical usefulness of the resulting interval.

Two researchers, Aoife and Cian, independently analyse the bimonthly electricity consumption (in kWh) of households in a district, whose distribution is normal with a known standard deviation $\sigma = 60\text{ kWh}$. Both use the exact same random sample of $n$ households. Aoife constructs a $95\%$ confidence interval ($z_{\alpha/2} = 1.96$) and obtains a total width of $23.52\text{ kWh}$. Cian decides to recalculate the interval with the same sample but increasing the confidence level to $99\%$ ($z_{\alpha/2} = 2.575$), and claims: "By increasing to $99\%$ confidence, our study gains reliability and precision, so the width of the interval will be smaller than Aoife's." a) Identify and refute Cian's error regarding the relationship between confidence and the width of an interval using mathematical arguments. b) Formulate the inverse problem: from the width of Aoife's interval ($23.52\text{ kWh}$), calculate the sample size $n$ that both used. c) If Cian wants to maintain his $99\%$ confidence level, but requires the width of his interval to be reduced to match Aoife's $23.52\text{ kWh}$, calculate how many additional households beyond those already surveyed he will need to collect.