Question

Question 4 2 pts What is the basic shape of the x2 - distribution? Skewed to the right Skewed to the left Bell-shaped O Uniform

Answer 1

4) The basic shape of a $\chi^2$ -distribution is

A) skewed to the right since one of the characteristics of a $\chi^2$ -distribution is that the $\chi^2$ -distribution is skewed to the right.

B)skewed to the left is false since as per the characteristics of $\chi^2$ -distribution the graph or basic shape of $\chi^2$ -distribution is skewed to the right.

C)bell-shaped is false because the characterisitcs of $\chi^2$ -distribution state that the basic shape of $\chi^2$ -distribution is skewed to the right but when df>90(degrees of freedom), the curve approximates to a normal distribution.

D)Uniform is false because the characteristics of $\chi^2$ -distribution state that the basic shape of $\chi^2$ -distribution is skewed to the right.

Hence the basic shape of a $\chi^2$ -distribution is skewed to the right.

5) We are given the following values

n=865(no.of voters surveyed), x=408(no. of voters favored approval of an issue before the legislature)

$p'=\frac{x}{n}=\frac{408}{865}=0.472$(0.4716763006 actually)

q'=1-p'=1-0.472=0.528

Since CL(confidence level) is 0.97, we know that $\alpha=1-0.97=0.03$ and

a = 0.015 2

The area to the right of z0.015 is 1-0.015=0.985.

From the areas of the standard normal distribution table when you look for the value 0.985, you obtain the critical value of 2.17.

Hence $z_\frac{\alpha}{2}=2.17$ (critical value of z at 3% level of significance)

The 97% confidence interval for true proportion is given by the formula,

$\left [ p'-z_\frac{\alpha}{2}\sqrt{\frac{p'q'}{n}} \leq p \leq p'+z_\frac{\alpha}{2}\sqrt{\frac{p'q'}{n}} \right ]$

Substituting the values in the above equation, we get the following confidence interval,

$\left [ 0.472-2.17\sqrt{\frac{0.472*0.528}{865}} \leq p \leq 0.472+2.17*\sqrt{\frac{0.472*0.528}{865}} \right ]$

$\Rightarrow \left [ 0.272,0.672 \right ]$ is the required confidence interval.

Hence 97% confidence interval for true proportion is,

0.272<p<0.672

7)

a)The original claim is that the standard deviation of hardness indexes for all such bolts is greater than 30 as evidenced by the line "Test the claim that the standard deviation of hardness indexes for all such bolts is greater than 30".

b) The null hypothesis is,

$H_0:\sigma^2=30$

i.e., the standard deviation of hardness indexes for all bolts is equal to 30.

c) The alternate hypothesis is,

$H_1:\sigma^2>30$

i.e., the standard deviation of hardness indexes for all bolts is greater than 30

d) We are given the following values and are asked to perform hypothesis test

Since this is a single variance hypothesis test, we use chi-square test for single variance test to conduct the hypothesis test.

n=12(no.of bolts), s=41.7(standard deviation of hardness index of all bolts), $\sigma=30$ (from null and alternate hypothesis).

Test statistic under H0

$\chi^2=\frac{(n-1)s^2}{\sigma^2}\sim \chi^2_{(n-1)}$

$=\frac{11*41.7^2}{30^2}$

=21.253(21.2531 actually)

Hence the test statistic for $\chi^2$ -test for single variance is 21.253

e) The p-value for $\chi^2$ -test statistic is 0.0308.