Step-by-step explanation:
Any number divided by zero, its value can't be determined. Hence it is called as indeterminate form. So is with zero divided by zero.
A.109 B.87 C.98 D.69
2 answers:
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The variable is
X: blood platelet count of a woman
This variable has a bell shaped distribution (Normal)
With mean μ=261.1 and standard deviation σ=64.3
a.
The platelet count within 3 standard deviations of the mean can be calculated as
μ±3σ
You can symbolize it as
μ-3σ ≤ X ≤ μ+3σ
According to the empirical rule of the normal distribution you know that under any bell shaped distribution:
Between μ ± σ you'll find the 68% of the distribution
Between μ ± 2σ you'll find the 95% of the distribution
Between μ ± 3σ you'll find the 99% of the distribution
So following this rule you'll find 99% of the women with platelet count within 3 standard deviations of the mean.
b.
3% of the population is found between a vertain interval, the rest 97% is separated in equal tails around this interval.
Divide 0.97 by 2
0.97/2= 0.485
Each tail contains 0.485 of the distribtuion.
a and b represent the values within you'll find 3% of the population.
Until a you'll find 0.485 of the population and until b you'll find 0.485+0.03=0.515
To calculate both unknown values you have to use the standard normal distribution. This distribution is centered in zero, the left tail is negative and the right tail is positive.
a and b are at a equal distance from the mean but with different sign, so you only have to calculate on and then you invert the sign to get the other one.
I'll calculate the positive value:
Z= (X- μ )/σ~N(0,1)
P(Z≤ b)=0.515
Look in the body of the Z-table, rigth or positive entry and reach the margins for the value:
b= 0.038
Then using the values of the mean and the standard deviation you can calculate the corresponding valueof platelet count:
b= (X- μ )/σ
0.038= (X- 261.1 )/64.3
0.038*64.3=X-261.1
2.4434+261.1=X
X=263.5434=263.5
a=-b=-0.038
a= (X- μ )/σ
-0.038= (X- 261.1 )/64.3
-0.038*64.3=X-261.1
-2.4434+261.1=X
X=258.6566= 258.7
3% of the populations platelet count is between 258.7 and 263.5.
Answer:
P(X ≤ 46 | X~B(821, 0.078)) = 0.00885745584
0.00885... < 0.01
The test statistic of 46 is significant
There is sufficient evidence to reject H₀ and accept H₁
Air bags are more effective as protection than safety belts
Step-by-step explanation:
821 crashes
46 hospitalisations where car has air bags
7.8% or 0.078 probability of hospitalisations in cars with automatic safety belts
α = 0.01 or 1% ← level of significance
One-tailed test
We are testing whether hospitalisations in cars with air bags are less likely than in a car with automatic safety belts;
The likelihood of hospitalisation in a car with automatic safety belts, we are told, is 7.8% or 0.078;
So we are testing if hospitalisations in cars with air bags is less than 0.078;
So, firstly:
Let X be the continuous random variable, the number of hospitalisations from a car crash with equipped air bags
X~B(821, 0.078)
Null hypothesis (H₀): p = 0.078
Alternative hypothesis (H₁): p < 0.078
According to the information, we reject H₀ if:
P(X ≤ 46 | X~B(821, 0.078)) < 0.01
To find P(X ≤ 46) or equally P(X < 47), it could be quite long-winded to do manually for this particular scenario;
If you are interested, the manual process involves using the formula for every value of x up to and including 46, i.e. x = 0, x = 1, x = 2, etc. until x = 46, the formula is:
You can find binomial distribution calculators online, where you input n (i.e. the number of trials or 821 in this case), probability (i.e. 0.078) and the test statistic (i.e. 46), it does it all for you, which gives:
P(X ≤ 46 | X~B(821, 0.078)) = 0.00885745584
Now, we need to consider if the condition for rejecting H₀ is met and recognise that:
0.00885... < 0.01
There is sufficient evidence to reject H₀ and accept H₁.
To explain what this means:
The test statistic of 46 is significant according to the 1% significance level, meaning the likelihood that only 46 hospitalisations are seen in car crashes with air bags in the car as compared to the expected number in car crashes with automatic safety belts is very unlikely, less than 1%, to be simply down to chance;
In other words, there is 99%+ probability that the lower number of hospitalisations in car crashes with air bags is due to some reason, such as air bags being more effective as a protective implement than the safety belts in car crashes.