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

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 $\bar x$.

Let $X\sim N(a,16)$. A sample of size $n=36$ yields a 95% confidence interval for $a$ of $(47,51)$. Find the sample mean $\bar x$.

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 $\mu$?

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 $\mu$.

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

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

Determine the minimum sample size $n$ required to estimate $\mu$ within $\pm3$ units using a 95% confidence interval if $\sigma=20$.

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

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 $\sigma$.

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 $b$.

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

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 $b$.

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

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

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.