Question

5.Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of nine months with a sample standard deviation of four months. Assume that the underlying population distribution is normal.
a. Construct a 99% confidence interval for the population mean length of time using training wheels.
(i) State the confidence interval
sketch the graph
calculate error bound

Answer 1

Let  the random variable $X$ be the time of using training wheelsin months

The provided sample mean is $\bar{X}=9$ and the sample standard deviation is $s=4.$ The size of the sample is $n=14$ and the required confidence level is 99%.

The number of degrees of freedom are $df=14-1=13,$ and the significance level is $\alpha=0.01.$

Based on the provided information, the critical t-value for $\alpha=0.01$ and $df=13$

degrees of freedom is $t_c=3.0123.$ The 95% confidence for the population $\mu$  is computed using the following expression

$$CI=(\bar{X}-\dfrac{t_c\times s}{\sqrt{n}}, \bar{X}+\dfrac{t_c\times s}{\sqrt{n}})=$$

$$=(9-\dfrac{3.0123\times 4}{\sqrt{14}}, 9+\dfrac{3.0123\times 4}{\sqrt{14}})=$$

$$=(5.78, 12.22)$$

$EBM=\dfrac{t_c\times s}{\sqrt{n}}=\dfrac{3.0123\times 4}{\sqrt{14}}=3.22$