Specialist Mathematics · Unit 4 · Statistical inference · Confidence intervals for means
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A quality control manager collects a random sample of 64 components from a production line. The masses of the components are normally distributed with unknown mean μ grams and known standard deviation σ = 12 grams. The sample mean is x̄ = 87.5 grams. (a) Calculate a 95% confidence interval for the population mean μ. (3 marks) (b) Explain what the confidence level of 95% means in the context of this interval estimate. (2 marks)
A researcher collected a random sample of \( n = 85 \) measurements of water temperature (in °C) from a large river. The sample mean is \( \bar{x} = 18.3 \)°C and the sample standard deviation is \( s = 2.1 \)°C. The researcher constructs a 95% confidence interval for the population mean temperature and obtains the interval \( [17.8, 18.8] \). Explain why this interval does not guarantee that the true population mean lies within these bounds.
A researcher measures the mass (in grams) of a random sample of 16 items from a production line. The sample mean is \(\bar{x} = 250.4\) g and the sample standard deviation is \(s = 2.8\) g. The population from which the sample is drawn is normally distributed. (a) Determine a 95% confidence interval for the population mean mass. (2 marks) (b) Evaluate whether the claim that the true population mean is 251.5 g is reasonable based on your interval estimate. (2 marks)
A quality-control manager measures the breaking strength (in newtons) of a random sample of 36 cables from a manufacturing batch. The sample mean is \(\bar{x} = 1,580\) N and the sample standard deviation is \(s = 120\) N. The manager constructs a 95% confidence interval for the mean breaking strength of the batch. For a sample of size 36, the critical value from the t-distribution at 95% confidence is \(t^* = 2.030\). (a) Determine the standard error of the mean. (1) (b) Find the margin of error for the confidence interval, correct to 1 decimal place. (1) (c) State the 95% confidence interval for the population mean breaking strength, correct to the nearest newton. (1) (d) Explain what is meant by the statement "95% confidence interval". (1)
A manufacturing plant measures the tensile strength of steel samples. A random sample of 36 specimens yields a mean tensile strength of 485 MPa with a sample standard deviation of 24 MPa. The plant manager claims that a 95% confidence interval for the true mean tensile strength lies between 477 and 493 MPa. (a) Determine the 95% confidence interval for the population mean tensile strength. (3 marks) (b) Evaluate the reasonableness of the manager's claim. (2 marks)
A quality-control manager at a manufacturing plant measures the tensile strength (in MPa) of steel rods from a large batch. A random sample of 36 rods is selected, and the sample mean is found to be \(\bar{x} = 485\) MPa with a sample standard deviation of \(s = 24\) MPa. The manager constructs a 95% confidence interval for the mean tensile strength of all rods in the batch using the formula \(\bar{x} \pm t^* \frac{s}{\sqrt{n}}\). (a) Calculate the margin of error for this confidence interval. (1) (b) State the 95% confidence interval for the mean tensile strength. (1) (c) Interpret what the 95% confidence level means in this context. (1) (d) Evaluate whether the current batch is likely to meet a quality standard that requires a mean tensile strength of at least 475 MPa. Justify your answer using the confidence interval. (1)
A veterinarian collects data on the resting heart rate (beats per minute) of a random sample of 36 adult dogs of the same breed. The sample yields a mean of 92 bpm with a sample standard deviation of 8 bpm. The veterinarian constructs a 95\% confidence interval for the population mean resting heart rate and claims it will capture the true population mean. Evaluate the reasonableness of this claim.
A random sample of 64 shipping containers from a large warehouse is measured, and the mean weight is found to be 1,240 kg with a sample standard deviation of 160 kg. A 95% confidence interval for the true population mean weight is calculated to be (1,200 kg, 1,280 kg). Explain why this interval estimate is more informative than simply reporting the sample mean of 1,240 kg alone.
A pharmaceutical company investigates the mean daily dosage level (in mg) of an active compound required to achieve a therapeutic effect. A random sample of 64 patients yields a sample mean of \(\bar{x} = 245\) mg with a known population standard deviation of \(\sigma = 32\) mg. (a) Calculate a 95% confidence interval for the population mean dosage. Give your answer correct to 1 decimal place. (2) (b) A quality assurance manager proposes that the true mean dosage lies between 238 mg and 252 mg with certainty. Evaluate the reasonableness of this proposal. (3)
A researcher estimates the mean lifespan of a particular species of insect using a random sample of 40 insects. The sample mean is 28.5 days with a sample standard deviation of 4.2 days. a) Explain why a point estimate of the population mean would be insufficient for the researcher's purposes. b) Construct a 95% confidence interval for the population mean lifespan, assuming the sample is approximately normally distributed. c) Interpret what the 95% confidence interval means in the context of this study.