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Nataly_w [17]
3 years ago
11

We use the t distribution for the statistical inference of the population mean when the underlying population standard deviation

is not known. Under the assumption that the population is normally distributed, find tα/2,df for the following scenarios. Use Table 2. (Round your answers to 3 decimal places.) tα/2,df
a. A 90% confidence level and a sample of 28 observations.
b. A 95% confidence level and a sample of 28 observations.
c. A 90% confidence level and a sample of 15 observations.
d. A 95% confidence level and a sample of 15 observations
Mathematics
1 answer:
Digiron [165]3 years ago
6 0

Answer:

Step-by-step explanation:

Since population std deviation is not  known, we use t critical value for testing of hypothesis.

a) 90% conf level when n =28

df = 27, t critical = 1.706

b) 95% 28

df =27 and t critical = 2.052

c) 90% and 15

df =14 and t critical = 1.761

d) 95% and n =15

df =14:   t critical = 2.145

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Answer:

(a) Decreasing on (0, 1) and increasing on (1, ∞)

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ƒ(x) = x² - x – lnx

(a) Intervals in which ƒ(x) is increasing and decreasing.

Step 1. Find the zeros of the first derivative of the function

ƒ'(x) = 2x – 1 - 1/x = 0

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We reject the negative root, because the argument of lnx cannot be negative.

There is one zero at (1, 0). This is your critical point.

Step 2. Apply the first derivative test.

Test all intervals to the left and to the right of the critical value to determine if the derivative is positive or negative.

(1) x = ½

ƒ'(½) = 2(½) - 1 - 1/(½) = 1 - 1 - 2 = -1

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(2) x = 2

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ƒ'(x) > 0 so the function is increasing on (1, ∞).

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ƒ(x) is decreasing when x < 1 and increasing when x >1.

Thus, (1, 0) is a local minimum, and ƒ(x) = 0 when x = 1.

(c) Inflection point

(1) Set the second derivative equal to zero

ƒ''(x) = 2 + 2/x² = 0

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                   x² = -2

There is no inflection point.

(2). Concavity

Apply the second derivative test on either side of the extremum.

\begin{array}{lccc}\text{Test} & x < 1 & x = 1 & x > 1\\\text{Sign of f''} & + & 0 & +\\\text{Concavity} & \text{up} & &\text{up}\\\end{array}

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