What is the equation of the line that passes through the point (6,-6)(6,−6) and has a slope of -\frac{5}{3}−

Step-by-step explanation:

a) We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean

For the null hypothesis,

µ = 35

For the alternative hypothesis,

µ > 35

It is a right tailed test

Since the number of samples is small and no population standard deviation is given, the distribution is a student's t.

Since n = 16,

Degrees of freedom, df = n - 1 = 16 - 1 = 15

t = (x - µ)/(s/√n)

Where

x = sample mean = 42.7

µ = population mean = 32

s = samples standard deviation = 6

t = (42.7 - 32)/(6/√16) = 7.13

We would determine the p value using the t test calculator. It becomes

p = 0.00001

Since alpha, 0.01 > than the p value, 0.00001, then we would reject the null hypothesis. Therefore, At a 1% level of significance, there is sufficient data to conclude that children with a history of child care show significantly more behavioral problems than the average kindergarten child

b) Confidence interval is written in the form,

(Sample mean - margin of error, sample mean + margin of error)

The sample mean, x is the point estimate for the population mean.

Margin of error = z × s/√n

the information given, the from population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score

In order to use the t distribution,

Since confidence level = 90% = 0.95, α = 1 - CL = 1 – 0.90 = 0.1

α/2 = 0.1/2 = 0.05

the area to the left of z0.05 is 0.05 and the area to the right of z0.05 is 1 - 0.05 = 0.95

Looking at the t distribution table for t.95 and df = 15

z = 1.753

Margin of error = 1.753 × 6/√16

= 2.63

Confidence interval = 35 ± 2.63