Question

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 six months with a sample standard deviation of

three months. Assume that the underlying population distribution is normal.

$$\begin{gathered} a.i.x̄=\_\_\_\_\_\_\_\_\_\_ \\ ii.s_x=\_\_\_\_\_\_\_\_\_\_ \\ iii.n=\_\_\_\_\_\_\_\_\_\_ \\ iv.n–1=\_\_\_\_\_\_\_\_\_\_ \end{gathered}$$

b. Define the random variable $X$in words.

c. Define the random variable $\overline{X}$in words.

d. Which distribution should you use for this problem? Explain your choice.

e. Construct a $99\%$confidence interval for the population mean length of time using training wheels.

i. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

f. Why would the error bound change if the confidence level were lowered to $90\%$?

Short answer

(a) The results are:

1. $x=6$

2. $s_x=3$

3. n = 14

4. n-1=13.

(b) The total amount of time that $1$youngster requires training wheels is $X$.

(c) From a sample of$14$youngsters, the average amount of time spent on training wheels is $\overline{X}$.

(d) With the parameters $t_{n-1}$, the distribution is the$t$.

$t_{14-1}=t_{13.}$

(e) The final results are:

1. $CI=(3.5848,8.4152)$

2. we show it through graph :

3.$EBM=1.72$

(f) The error bound will become smaller as the confidence level decreases, because the area under the curve to derive the real population mean decreases as the confidence level decreases.

Step-by-step solution

Explanation (a)

i. The sample mean time spent utilising training wheels is $6$minutes.

Overline $x=6$is the mean time.

ii. The sample length of time's standard deviation, $s_x=3$.

iv. There were $14$children asked about how long they had been wearing training wheels, $n=14.$

iv. If $n=14$is the total number of patients used in the study, then $n-1=13$.

Explanation (b)

The total amount of time that $1$ youngster requires training wheels is $X$.

Explanation (c)

From a sample of $14$youngsters, the average amount of time spent on training wheels is $\overline{X}$.

Explanation (d)

With the parameters $t_{n-1}$, the distribution is the t.

$t_{14-1}=t_{13.}$

Explanation (e) 

i. Using the $TI-83$calculator, calculate the confidence interval. The confidence interval's output,

$CI=(3.5848,8.4152)$

ii. The graph is as follows:

iii. The formula is used to compute the error bound,

$EBM=t_{n-1}\left(\frac{\alpha }{2}\right)\frac{s}{\sqrt{n}}$

$EBM=t_{14-1}\left(\frac{0.05}{2}\right)\frac{3}{\sqrt{14}}$

$EBM=1.72$.

Explanation (f)

The error bound will become smaller as the confidence level decreases, because the area under the curve to derive the real population mean decreases as the confidence level decreases.