Help?! No clue about functions...
2 answers:
Method 1: Using Calculus
The first method in going about this is to use Calculus, a branch of mathematics interested in the rate of change of a certain graph. In order to use Calculus, we need to understand what we are required to do when finding minimum values.
We first need to take the derivative of the function, because this becomes our tangent or slope at any arbitrary point on the graph.
We now need the derivative to be equal to zero, as this is will give us a horizontal tangent, which means we are able to find the turning point of a graph.
So, at x = 4, there will be a minimum, since the second derivative is clearly greater than zero. By substituting this point back to f(x), we can find a value for y, and this is our minimum point of the curve.
Thus, we know the minimum point of this curve, f(x), is at: (4, 5)
Method 2: Vertex form
For any given quadratic, or a polynomial with degree 2, 4, 6, etc., we can easily find the minimum point, as that simply is our vertex. By converting this into vertex form, we can find the coordinates that will satisfy the vertex of this function, which becomes our minimum point.
Start by completing the square:
Thus, the vertex becomes: (4, 5) which is our lowest point (ie our minimum point)
The first method in going about this is to use Calculus, a branch of mathematics interested in the rate of change of a certain graph. In order to use Calculus, we need to understand what we are required to do when finding minimum values.
We first need to take the derivative of the function, because this becomes our tangent or slope at any arbitrary point on the graph.
We now need the derivative to be equal to zero, as this is will give us a horizontal tangent, which means we are able to find the turning point of a graph.
So, at x = 4, there will be a minimum, since the second derivative is clearly greater than zero. By substituting this point back to f(x), we can find a value for y, and this is our minimum point of the curve.
Thus, we know the minimum point of this curve, f(x), is at: (4, 5)
Method 2: Vertex form
For any given quadratic, or a polynomial with degree 2, 4, 6, etc., we can easily find the minimum point, as that simply is our vertex. By converting this into vertex form, we can find the coordinates that will satisfy the vertex of this function, which becomes our minimum point.
Start by completing the square:
Thus, the vertex becomes: (4, 5) which is our lowest point (ie our minimum point)
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Answer:
a) [ -27.208 , -12.192 ]
b) New procedure is not recommended
Step-by-step explanation:
Solution:-
- It is much more common for a statistical analyst to be interested in the difference between means than in the specific values of the means themselves.
- The principal at Crest Middle School collects data on how much time students at that school spend on homework each night.
- He/She takes a " random " sample of n = 20 from a sixth and seventh grades students from the school population to conduct a statistical analysis.
- The summary of sample mean ( x1 & x2 ) and sample standard deviation ( s1 & s2 ) of the amount of time spent on homework each night (in minutes) for each grade of students is given below:
<u>Mean ( xi )</u> <u> Standard deviation ( si )</u>
Sixth grade students 27.3 10.8
Seventh grade students 47.0 12.4
- We will first check the normality of sample distributions.
- We see that sample are "randomly" selected.
- The mean times are independent for each group
- The groups are selected independent " sixth " and " seventh" grades.
- The means of both groups are conforms to 10% condition of normality.
Hence, we will assume that the samples are normally distributed.
- We are to construct a 95% confidence interval for the difference in means ( u1 and u2 ).
- Under the assumption of normality we have the following assumptions for difference in mean of independent populations:
- Population mean of 6th grade ( u1 ) ≈ sample mean of 6th grade ( x1 )
- Population mean of 7th grade ( u2 ) ≈ sample mean of 6th grade ( x2 )
Therefore, the difference in population mean has the following mean ( u ^ ):
u^ = u1 - u2 = x1 - x2
u^ = 27.3 - 47.0
u^ = -19.7
- Similarly, we will estimate the standard deviation (Standard Error) for a population ( σ^ ) represented by difference in mean. The appropriate relation for point estimation of standard deviation of difference in means is given below:
σ^ = √ [ ( σ1 ^2 / n1 ) + ( σ2 ^2 / n2 ) ]
Where,
σ1 ^2 : The population variance for sixth grade student.
σ2 ^2 : The population variance for sixth grade student.
n1 = n2 = n : The sample size taken from both populations.
Therefore,
σ^ = √ [ ( 2*σ1 ^2 / n )].
- Here we will assume equal population variances : σ1 ≈ σ2 ≈ σ is "unknown". We can reasonably assume the variation in students in general for the different grade remains somewhat constant owing to other reasons and the same pattern is observed across.
- The estimated standard deviation ( σ^ ) of difference in means is given by:
σ^ =
σ^ = 11.62755*√2/20
σ^ = 3.67695
- Now we will determine the critical value associated with Confidence interval ( CI ) which is defined by the standard probability of significance level ( α ). Such that:
Significance Level ( α ) = 1 - CI = 1 - 0.95 = 0.05
- The reasonable distribution ( T or Z ) would be determined on the basis of following conditions:
- The population variances ( σ1 ≈ σ2 ≈ σ ) are unknown.
- The sample sizes ( n1 & n2 ) are < 30.
Hence, the above two conditions specify the use of T distribution critical value. The degree of freedom ( v ) for the given statistics is given by:
v = n1 + n2 - 2 = 2n - 2 = 2*20 - 2
v = 38 degrees of freedom
- The t-critical value is defined by the half of significance level ( α / 2 ) and degree of freedom ( v ) as follows:
t-critical = t_α / 2, v = t_0.025,38 = 2.024
- Then construct the interval for 95% confidence as follows:
[ u^ - t-critical*σ^ , u^ + t-critical*σ^ ]
[ -19.7 - 2.042*3.67695 , -19.7 + 2.042*3.67695 ]
[ -19.7 - 7.5083319 , -19.7 + 7.5083319 ]
[ -27.208 , -12.192 ]
- The principal should be 95% confident that the difference in mean times spent of homework for ALL 6th and 7th grade students in this school (population) lies between: [ -27.208 , -12.192 ]
- The procedure that the matched-pairs confidence interval for the mean difference in time spent on homework prescribes the integration of time across different sample groups.
- If we integrate the times of students of different grades we would have to make further assumptions like:
- The intelligence levels of different grade students are same
- The aptitude of students from different grades are the same
- The efficiency of different grades are the same.
- We have to see that both samples are inherently different and must be treated as separate independent groups. Therefore, the above added assumptions are not justified to be used for the given statistics. The procedure would be more bias; hence, not recommended.