4.16 Confidence intervalsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The masses of cats of a certain breed are normally distributed with a known standard deviation of $0.6$ kg. A random sample of $25$ of these cats has a mean mass of $4.20$ kg. A confidence interval is to be found for the population mean mass $\mu$.

   (a)

   Which of the following should be used to construct a $95\%$ confidence interval for $\mu$?

   [1 mark]

   • Athe normal distribution, with $\sigma=0.6$
   • Bthe $t$-distribution with $24$ degrees of freedom
   • Cthe $t$-distribution with $25$ degrees of freedom
   • Dthe binomial distribution with $n=25$

   (b)

   Use your GDC to find a $95\%$ confidence interval for $\mu$.

   [1 mark]

   • A$(4.00,\ 4.40)$
   • B$(3.02,\ 5.38)$
   • C$(4.06,\ 4.34)$
   • D$(3.96,\ 4.44)$

   (c)

   The breed society claims that the mean mass of these cats is $4.5$ kg. Use your interval from (b) to comment on this claim.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The lifetimes, in hours, of a type of battery are normally distributed with unknown mean $\mu$ and unknown standard deviation. The lifetimes of a random sample of $12$ batteries are: $40.2,\ 38.9,\ 41.5,\ 39.8,\ 42.1,\ 40.7,\ 39.3,\ 41.0,\ 40.4,\ 38.6,\ 41.8,\ 40.2$. Use your GDC where helpful.

   (a)

   Which critical value is used to construct a $95\%$ confidence interval for $\mu$?

   [1 mark]

   • A$1.96$
   • B$2.20$
   • C$2.18$
   • D$1.80$

   (b)

   Use your GDC to find an unbiased estimate of the population standard deviation, in hours.

   [1 mark]

   • A$1.07$
   • B$1.24$
   • C$1.11$
   • D$0.321$

   (c)

   Use your GDC to find a $95\%$ confidence interval for $\mu$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The daily screen time of teenagers, in minutes, is normally distributed with a known standard deviation. A $95\%$ confidence interval for the population mean $\mu$, based on a random sample of $64$ teenagers, is $(152.65,\ 167.35)$.

   (a)

   (i) Write down the sample mean.
   (ii) Find the value of the population standard deviation.

   [3 marks]

   (b)

   Using the same sample, find a $90\%$ confidence interval for $\mu$. State, with a reason, whether it is narrower or wider than the $95\%$ interval.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The service times, in minutes, at two cafés, A and B, are each normally distributed.

   (a)

   A random sample of $15$ customers at café A has a mean service time of $4.80$ minutes and an unbiased estimate of the population standard deviation of $1.20$ minutes.
   (i) Explain why the $t$-distribution is used to find a confidence interval for the mean service time at café A.
   (ii) Find a $95\%$ confidence interval for this mean.
   (iii) The owner of café A claims that the mean service time is $5$ minutes. Comment on this claim.

   [6 marks]

   (b)

   At café B the population standard deviation is known to be $1.1$ minutes. A random sample of $40$ customers has a mean service time of $4.5$ minutes.
   (i) Find a $99\%$ confidence interval for the mean service time at café B.
   (ii) Find the smallest sample size needed for the margin of error of a $99\%$ confidence interval to be at most $0.2$ minutes.

   [6 marks]

   Total for question 4: 12 marks

End of questions