Exercises for Question Type 3: Solving for the value of some parameters given the confidence interval - IB | RevisionDojo

Question Type 3: Solving for the value of some parameters given the confidence interval Exercises

Question 1

Let $X \sim N(\mu, 9)$ . A random sample of size $n = 25$ yields sample mean $\bar{x} = 50$ . Calculate a $95\%$ confidence interval for $\mu$ .

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Question 2

Determine the minimum sample size $n$ required to estimate the population mean $\mu$ within $\pm3$ units, using a 95% confidence interval, given that the population standard deviation is $\sigma=20$ .

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Question 3

Let $X \sim N(a, 16)$ and $n = 16$ . A confidence interval for $a$ is given by $(\bar{x} - 2, \bar{x} + 2)$ . Find its confidence level.

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Question 4

This question assesses the ability to determine the variance of a normal distribution given a confidence interval for the mean when the variance is known.

Let $X \sim N(a, b)$ with $b$ known. A sample of size $n = 20$ gives a $95\%$ confidence interval for $a$ of $(3, 6)$ . Find the value of $b$ .

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Question 5

Topic: Statistics and Probability

A manufacturer claims that the average lifetime of a product is $\mu$ hours with known $\sigma = 50$ hours. How many observations are needed to be $95\%$ confident that the sample mean is within $10$ hours of $\mu$ ?

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Question 6

Let $X \sim N(a, 9)$ . A sample of size $n = 36$ gives $\bar{x} = 15$ . Construct a 98% confidence interval for $a$ .

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Question 7

A $95\%$ confidence interval for the mean $a$ of a normal population is given as $(47, 51)$ .

Find the sample mean, $\bar{x}$ .

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Question 8

Determine the sample size needed to achieve a margin of error of $1$ unit for a $99\%$ confidence interval if $\sigma = 10$ .

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Question 9

Confidence Intervals for a Population Mean

Determine the 95% confidence interval for $\mu$ given a random sample of size $n = 49$ from $N(\mu, 36)$ , with sample mean $\bar{x} = 120$ .

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Question 10

The normal distribution and confidence intervals for the mean.

Suppose $X\sim N(a,b)$ . A sample of size $n=100$ gives $\bar{x}=30$ . The $99\%$ confidence interval for $a$ is $30\pm1$ . Find the value of $b$ .

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Question 11

Let $X \sim N(\mu, 4)$ and $n = 64$ . The sample mean is $\bar{x} = 20$ . Determine the 90% confidence interval for $\mu$ .

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Question 12

Let $X \sim N(a, b)$ . A 90% confidence interval for $a$ based on $n = 50$ is $(12.4, 13.6)$ . Find the value of $b$ .

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Question 13

This question requires the calculation of the sample mean from a given symmetric confidence interval for a normally distributed population.

Let $X \sim N(a, 25)$ . A sample of size $n = 25$ yields a $95\%$ confidence interval for $a$ of $(48, 52)$ .

Find the sample mean $\bar{x}$ .

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Question 14

Let $X \sim N(a, b)$ . For a sample of size $n = 30$ , a $99\%$ confidence interval for $a$ is $(10, 14)$ . Determine the value of $b$ .

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Question 15

A 95% confidence interval for $\mu$ is observed to have total width 10. If $n = 25$ and $X \sim N(\mu, \sigma^2)$ with known $\sigma$ , find $\sigma$ .

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