Asample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.48 mpa and a sample standard deviation of .79 mpa ("characterization of bearing strength factors in pegged timber connections, "j. of structural engr., 1997: 326-332). assume the population has normal distribution. calculate a 95% lower confidence bound for the true average proportional limit stress of all such joints. calculate a 95% confidence interval for the true average proportional limit stress of all such joints.

lower bound-8.106

8.02≤μ≤8.93 is the confidence interval

Step-by-step explanation:

given

mean(x)=8.48 MPa

standard deviation(s)=0.79 MPa

as we dont know the population standard deviation  so we use t-stat

formula

tα,n-1=t=}" alt="\frac{s}{√n}" />}" />

where

s-sample standard deviation

x-sample mean

μ-population mean

n-sample size

for 95% confidence interval and 13 degrees of freedom

t=1.771 (one tail ,as only lower bound is needed)

for lower bound

x-t×≤μ

8.48-1.771×≤μ

μ≥8.106

confidence interval

t=2.160 (two tailed)

x-t×≤μ≤x+t×

8.48-2.160×≤μ≤8.48+2.160×

8.02≤μ≤8.93 is the confidence interval required

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is