IB Mathematics Applications & Interpretation Question Type 3: Solving for the Value of Some Parameters Given the Confidence Interval Exercises

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 $\mu$.

Determine the minimum sample size $n$ required to estimate the population mean $\mu$ within $\pm 3$ units, using a 95% confidence interval, given that the population standard deviation is $\sigma =20$.

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.

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 $b$.

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

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

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

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

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

Suppose $X\sim N(a,b)$. A sample of size $n=100$ gives $\bar{x}=30$. The $99\%$ confidence interval for $a$ is $30\pm 1$. Find the value of $b$.

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

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 $b$.

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

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 $\sigma$.