Moderate -0.3 This is a standard normal distribution question requiring routine z-score calculations and inverse normal lookups. Part (a) involves straightforward probability-to-frequency conversion, part (b) uses symmetry and tables, and part (c) requires solving simultaneous equations from two z-scores—all well-practiced S1 techniques with no novel problem-solving required.
Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
Find the expected number of days during a year (52 weeks) for which Zak takes less than 30 minutes for his run. [4]
The probability that Zak's time is between 35.2 minutes and \(t\) minutes, where \(t > 35.2\), is 0.148. Find the value of \(t\). [3]
The random variable \(X\) has the distribution \(\text{N}(\mu, \sigma^2)\). It is given that \(\text{P}(X < 7) = 0.2119\) and \(\text{P}(X < 10) = 0.6700\). Find the values of \(\mu\) and \(\sigma\). [5]
\begin{enumerate}[label=(\alph*)]
\item Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
\begin{enumerate}[label=(\roman*)]
\item Find the expected number of days during a year (52 weeks) for which Zak takes less than 30 minutes for his run. [4]
\item The probability that Zak's time is between 35.2 minutes and $t$ minutes, where $t > 35.2$, is 0.148. Find the value of $t$. [3]
\end{enumerate}
\item The random variable $X$ has the distribution $\text{N}(\mu, \sigma^2)$. It is given that $\text{P}(X < 7) = 0.2119$ and $\text{P}(X < 10) = 0.6700$. Find the values of $\mu$ and $\sigma$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2015 Q7 [12]}}