CI with two different confidence levels same sample

CAIE FP2 2009 June Q6

6 marks Standard +0.8

6 The times taken by employees in a factory to complete a certain task have a normal distribution with mean $\mu$ seconds and standard deviation $\sigma$ seconds, both of which are unknown. Based on a random sample of 20 employees, the symmetric $95 \%$ confidence interval for $\mu$ is $( 481,509 )$. Calculate a symmetric $90 \%$ confidence interval for $\mu$.
[0pt] [6]

Edexcel S3 2024 January Q6

15 marks Standard +0.3

A random sample of 8 three-month-old golden retriever dogs is taken.

The heights of the golden retrievers are recorded.
Using this sample, a 95\% confidence interval for the mean height, in cm, of three-month-old golden retrievers is found to be $( 45.72,53.88 )$

Find a 99\% confidence interval for the mean height. You may assume that the heights are normally distributed with known population standard deviation. Some summary statistics for the weights, $x \mathrm {~kg}$, of this sample are given below. $$\sum x = 91.2 \quad \sum x ^ { 2 } = 1145.16 \quad n = 8$$
Calculate unbiased estimates of the mean and the variance of the weights of three-month-old golden retrievers. A further random sample of 24 three-month-old golden retrievers is taken. The unbiased estimates of the mean and the variance of the weights, in kg , from this sample are found to be 10.8 and 17.64 respectively.
Estimate the standard error of the mean weight for the combined sample of 32 three-month-old golden retrievers.

Edexcel S3 2018 June Q5

7 marks Standard +0.3

5. A factory produces steel sheets whose weights, $X \mathrm {~kg}$, have a normal distribution with an unknown mean $\mu \mathrm { kg }$ and known standard deviation $\sigma \mathrm { kg }$. A random sample of 25 sheets gave both a

$95 \%$ confidence interval for $\mu$ of $( 30.612,31.788 )$
$c \%$ confidence interval for $\mu$ of $( 30.66,31.74 )$
1. Find the value of $\sigma$
2. Find the value of $c$, giving your answer correct to 3 significant figures.

Edexcel S3 2018 Specimen Q8

9 marks Standard +0.3

8. A factory produces steel sheets whose weights $X \mathrm {~kg}$, are such that $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to be (29.74, 31.86)

Find, to 2 decimal places, the standard error of the mean.
Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample of sheets. Using four different random samples, four $90 \%$ confidence intervals for $\mu$ are to be found.
Calculate the probability that at least 3 of these intervals will contain $\mu$. \section*{8. A factory produces steel sheets whose weights $X \mathrm { gg }$, are such $X \sim N ( \mu , \sigma ) ^ { 2 }$} A. A. A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to
be $( 29.74,31.86 )$
1. Find, to 2 decimal places, the standard error of the mean.
2. Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample
  of sheets. (3)
  \includegraphics[max width=\textwidth, alt={}]{0434a6c1-686a-449d-ba16-dbb8e60288e8-28_2646_1824_105_123}
  
  VIIIV SIHI NI JIIIM ION OC VIUV SIHI NI JIIAM ION OO VEXV SIHI NI JIIIM IONOO
  
  VIIIV SIHI NI IIIYM ION OC VIIV SIHI NI JIIIM ION OC VI4V SIHI NI JIIIM I ION OO

Edexcel S3 2007 June Q6

6 marks Standard +0.3

A random sample of the daily sales (in £s) of a small company is taken and, using tables of the normal distribution, a 99\% confidence interval for the mean daily sales is found to be
(123.5, 154.7)

Find a $95 \%$ confidence interval for the mean daily sales of the company.
(6)

Edexcel S3 2013 June Q5

9 marks Standard +0.3

A manufacturer produces circular discs with diameter $D \mathrm {~mm}$, such that $D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$. A random sample of discs is taken and, using tables of the normal distribution, a $90 \%$ confidence interval for $\mu$ is found to be
(118.8, 121.2)
1. Find a 98\% confidence interval for $\mu$.
2. Hence write down a 98\% confidence interval for the circumference of the discs.
Using three different random samples, three $98 \%$ confidence intervals for $\mu$ are to be found.
Calculate the probability that all the intervals will contain $\mu$.

Edexcel S3 2015 June Q8

9 marks Standard +0.3

A factory produces steel sheets whose weights $X$ kg, are such that $X \sim \text{N}(\mu, \sigma^2)$ A random sample of these sheets is taken and a 95\% confidence interval for $\mu$ is found to be (29.74, 31.86)

Find, to 2 decimal places, the standard error of the mean. [3]
Hence, or otherwise, find a 90\% confidence interval for $\mu$ based on the same sample of sheets. [3]

Using four different random samples, four 90\% confidence intervals for $\mu$ are to be found.

Calculate the probability that at least 3 of these intervals will contain $\mu$. [3]