Question

A random sample of 36 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 11 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 10.5.

(a) Is it appropriate to use a Student's *t*
distribution? Explain.

Yes, because the *x* distribution is mound-shaped and
symmetric and σ is unknown.No, the *x* distribution is
skewed left. No, the *x*
distribution is skewed right.No, the *x* distribution is not
symmetric.No, σ is known.

How many degrees of freedom do we use?

(b) What are the hypotheses?

*H*_{0}: μ = 10.5; *H*_{1}: μ >
10.5*H*_{0}: μ = 10.5; *H*_{1}: μ ≠
10.5 *H*_{0}: μ <
10.5; *H*_{1}: μ = 10.5*H*_{0}: μ =
10.5; *H*_{1}: μ < 10.5*H*_{0}: μ
> 10.5; *H*_{1}: μ = 10.5

(c) Compute the *t* value of the sample test statistic.
(Round your answer to three decimal places.)

t =

(d) Estimate the *P*-value for the test.

*P*-value > 0.250

0.100 < *P*-value <
0.250

0.050 < *P*-value < 0.100

0.010 < *P*-value < 0.050

*P*-value < 0.010

(e) Do we reject or fail to reject *H*_{0}?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(f) Interpret the results.

There is sufficient evidence at the 0.05 level to reject the null hypothesis. OR

There is insufficient evidence at the 0.05 level to reject the null hypothesis.

Answer #1

a)

Yes, because the *x* distribution is mound-shaped and
symmetric and σ is unknown

b)

Define,

Here

Here we test

Then the test-statistic

which is distributed as a t with df= n-1

i.e. degrees of freedom

Here,

So,

d)

e)

At the level, we fail to reject the null hypothesis and conclude the data are statistically significant.

f)

There is insufficient evidence at the 0.05 level to reject the null hypothesis.

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