IB Mathematics Analysis and Approaches Topic SL…

A clinic measures the systolic blood pressure of all visiting patients. The blood pressure $P$ , in mmHg, of patients is normally distributed with a mean of $122$ mmHg and a standard deviation of $9$ mmHg. A patient is classified as having elevated blood pressure if $P > 130$ mmHg.

A patient is selected at random.

Assume that the blood pressure values of randomly selected patients are independent.

(a) Find the probability that this patient has elevated blood pressure.

(b) Given that the patient has elevated blood pressure, find the probability that their blood pressure is greater than $140$ mmHg.

(d) Three hundred patients are selected at random.

Find the probability that at least $50$ of these $300$ patients have elevated blood pressure.

(e) Three hundred patients are selected at random.

Find the expected number of patients with elevated blood pressure among these $300$ patients.

A bakery produces loaves of bread. The weights $X$ , in grams, of loaves can be modelled by a normal distribution where $X \sim N(\mu, \sigma^2)$ . A loaf is classified as substandard if it weighs less than $480$ grams.

(a) Given that $\mu = 510$ and $\sigma = 12$ , find the probability that a randomly selected loaf is substandard.

Let $W$ be the mass, in g, of packets filled by a machine. The masses are normally distributed with mean $\mu=250\text{ g}$ and variance $\sigma^2=25\text{ g}^2$ .

(b) Find the value of $k$ (in g) such that $P(250-k<W<250+k)=0.70$ .

Let $H$ be the height, in cm, of a student in a school. The heights are normally distributed with mean $\mu=165\mathrm{~cm}$ and variance $\sigma^{2}=64\mathrm{~cm}^{2}$ .

(b) Find the value of $k$ (in cm) such that $P(165-k<H<165+k)=0.60$ .

Let $X$ and $Y$ be normally distributed with $X \sim N(80, \sigma^2)$ and $Y \sim N(56, \sigma^2)$ , where the values are measured in kilograms.

(a) Find $k$ such that $P(Y > k) = P(X < k)$ .

(b) Given that $P(X > 76) = 0.7881$ , find $P(60 < Y < 65)$ .

(b) The bakery decides that the probability that a loaf is substandard should be $0.0008$ . To do this $\mu$ is increased and $\sigma$ is kept unchanged.

the new value of $\mu$ , rounding your answer to the nearest gram.

(c) The bakery is happy with the decision that the probability that a loaf is substandard should be $0.0008$ , but is unhappy with the way in which this was achieved. The machines are now adjusted to reduce $\sigma$ and return $\mu$ back to $510$ grams.

the new value of $\sigma$ , rounding your answer to the nearest gram.

IB Mathematics Analysis and Approaches SL 4.12—Z Values, Inverse Normal to Find Mean and Standard Deviation Questionbank

IB Mathematics Analysis and Approaches SL 4.12—Z Values, Inverse Normal to Find Mean and Standard Deviation Questionbank

IB Mathematics Analysis and Approaches SL 4.12—Z Values, Inverse Normal to Find Mean and Standard Deviation Questionbank

IB Mathematics Analysis and Approaches SL 4.12—Z Values, Inverse Normal to Find Mean and Standard Deviation Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 4.12—Z Values, Inverse Normal to Find Mean and Standard Deviation Questionbank