The time (in number of days) until maturity of a certain variety of tomato plant is normally distributed with mean µ and standard deviation σ = 2.4. If you select a simple random sample of four plants of this variety and measure the time until maturity. The four times, in days are: 63, 69, 62, 66. If I wanted the margin of error for the 99% confidence interval to be ± 1 inch, I should select a simple random sample of size

a. Based on these data, a 99% confidence interval for μ , in days, is .
b. If I wanted the margin of error for the 99% confidence interval to be ± 1 inch, I should select a simple random sample of size.

a)    

So on this case the 99% confidence interval would be given by (60.404;66.596)    

b)

So the answer for this case would be n=39 rounded up to the nearest integer

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

represent the sample mean for the sample  

population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Part a

The confidence interval for the mean is given by the following formula:

  (1)

In order to calculate the mean  we can use the following formula:  

(2)  

The mean calculated for this case is

Since the Confidence is 0.99 or 99%, the value of and , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that

Now we have everything in order to replace into formula (1):

So on this case the 99% confidence interval would be given by (60.404;66.596)    

Part b

The margin of error is given by this formula:

   (a)

And on this case we have that ME =1 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

  (b)

The critical value for 99% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.005;0;1)", and we got , replacing into formula (b) we got:

So the answer for this case would be n=39 rounded up to the nearest integer

the image is given below

since the options are not given so we draw it from the equation

the given equation is

y = (x-1) (x+4)

y = y² + 3x - 4

the above equation is a quadratic equation which will have two roots. if we draw the graph for it, it will look like this

2y+y=5

add 2y and y