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Quiz 9: t-tables; Z-tests and t-tests for a mean

Question

Question 1

Use tables to find an interval containing P(t₆ > 1.5).

Not correct. Choice (a) is false.

Draw a diagram and mark in 1.5 on the x-axis. From the tables, P(t₆ > 1.943) = 0.05. Now mark in 1.943 on the x-axis, and it will be obvious that P(t₆ > 1.5) must be greater than 0.05.

Your answer is correct.

Not correct. Choice (c) is false.

Draw a diagram and mark in 1.5 on the x-axis. From the tables, P(t₆ > 1.440) = 0.10. Now mark in 1.44 on the x-axis, and it will be obvious that P(t₆ > 1.5) must be smaller than 0.10.

Not correct. Choice (d) is false.

Try again.

Question 2

Let τ = ¯
XS∕-√μn , where X and S are variables representing the mean and standard deviation of a sample of size n = 13 from the normal (μ,σ²) population. From the tables, the value of c such that P(τ > c) = 0.05 is

Not correct. Choice (a) is false.

Try again. τ does not have the t₁₃ distribution.

Not correct. Choice (b) is false.

Try again. τ = -¯X-√μ-
S∕ 13

is not normally distributed.

Not correct. Choice (c) is false.

What is the distribution of τ = -¯X-μ-
S∕√13

Your answer is correct.

τ =

~ t₁₂ and P(t₁₂ > 1.782) = 0.05.

Question 3

Your answer is correct.

This an upper-tail test, as the anticipated alternative hypothesis is a mean greater than 5 units.

Not correct. Choice (b) is false.

Try again. What is the anticipated alternative?

Not correct. Choice (c) is false.

Try again. x is the observed mean.

Not correct. Choice (d) is false.

Try again. This is not a two-tailed test.

Question 4

Questions 3 - 5 use the following information.
The mean fineness μ, of yarn is expected to be greater than the standard value of 5 units. To test this claim, the factory graded 16 specimens of the yarn and found the sample average, x, to be 5.9 units.
Assume that the measurement of fineness is normally distributed with a variance of σ² = 4.0. We can state:

Your answer is correct.

Since σ = 2, and this is an upper-tail test, the observed value of the Z-statistic is 52.9∕√-156-

= 1.8 with P-value = P(Z ≥ 1.8) = 0.0359.

Not correct. Choice (b) is false.

Try again. The observed value of the Z-statistic is not 0.45.

Not correct. Choice (c) is false.

Try again. The observed value of the Z-statistic is not 3.6.

Not correct. Choice (d) is false.

Try again. This is not a two-tailed test.

Question 5

Questions 3 - 5 use the following information.
The mean fineness μ, of yarn is expected to be greater than the standard value of 5 units. To test this claim, the factory graded 16 specimens of the yarn and found the sample average, x, to be 5.9 units. Assume that the measurement of fineness is normally distributed with unknown variance. State the distribution of the appropriate test statistic, τ, and find the P-value if the standard deviation of the sample is s = 2.4.

Not correct. Choice (a) is false.

Try again. The test statistic is not normal when σ is unknown.

Not correct. Choice (b) is false.

Try again. You have the wrong t distribution.

Not correct. Choice (c) is false.

Try again. You may have misread the t-tables.

Your answer is correct.

τ_obs =

= 1.5 and P(t₁₅ > 1.5) is in the interval (0.05, 0.10).

Question 6

Questions 6 - 9 use the following information. A manufacturing process produces components which have weight normally distributed about 60g with a standard deviation of 1.2g. A new process has been developed to cut costs. Below are the weights of a random sample of 15 components produced by the new process:

60.3	59.8	62.5	60.8	61.6	59.9	61.2	59.4
61.0	58.9	62.1	60.7	59.1	60.2	63.1

The management wishes to know if the new process results in a different mean weight. Which of the following are the most appropriate null and alternative hypotheses?

Not correct. Choice (a) is false.

Try again. There is no supporting information to anticipate an increase in weight.

Your answer is correct.

We perform a two-sided test as there is no anticipated direction for the alternative hypothesis.

Not correct. Choice (c) is false.

Try again. Think about the value of $μ$ and what sort of test we need to perform.

Not correct. Choice (d) is false.

Try again. Think about the value of $μ.$

Question 7

60.3	59.8	62.5	60.8	61.6	59.9	61.2	59.4
61.0	58.9	62.1	60.7	59.1	60.2	63.1

The management wishes to know if the new process results in a different mean weight. Assuming the standard deviation is unchanged for the new process, which of the following is the most appropriate test statistic, τ?

Not correct. Choice (a) is false.

Try again, we have a sample of 15 measurements.

Your answer is correct.

Since $σ$ is known, we know that $-- 2 2 X ~ N (μ, σn-) ~ N (60, 1.125-)$ . Standardising X gives the test statistic, τ = --
1X.2-∕6√015

(0,1)

Not correct. Choice (c) is false.

You have not standardised X correctly.

Not correct. Choice (d) is false.

Try again, $σ$ is known.

Question 8

60.3	59.8	62.5	60.8	61.6	59.9	61.2	59.4
61.0	58.9	62.1	60.7	59.1	60.2	63.1

The management wishes to know if the new process results in a different mean weight. State the P-value of the Z-test.

Not correct. Choice (a) is false.

Try again, remember we are performing a two-sided test.

Not correct. Choice (b) is false.

Try again.

Your answer is correct.

Large and small values of $-- X$ argue against $H0$ .
The observed value of $-- X$ is 60.71.
$P (|Z| > 60.71-√60) = P (|Z | > 2.29) = 2(1- Φ (2.29)) = 2(1- 0.9890) = 0.022. 1.2∕ 15$

Not correct. Choice (d) is false.

Check that the sample mean is 60.71 and recalculate the observed value of the Z-statistic.

Question 9

60.3	59.8	62.5	60.8	61.6	59.9	61.2	59.4
61.0	58.9	62.1	60.7	59.1	60.2	63.1

The management wishes to know if the new process results in a different mean weight. If we are not prepared to assume that the standard deviation is 1.2g for the new process, find τ_obs, the observed value of the appropriate t-statistic, and provide a P-value.

Not correct. Choice (a) is false.

Try again. You need to calculate both x and s and substitute into τ_obs = -¯x-√60-
s∕ 15

Not correct. Choice (b) is false.

Try again. Check your calculation of the sample standard deviation, s = 1.24, and substitute x and s into τ_obs = -¯x-√60-
s∕ 15

Not correct. Choice (c) is false.

Try again. This is not an upper-tail test.

Your answer is correct.

Since σ is unknown, τ_obs = x-μ-
s∕√n-

= 2.22
and P-value = P(|t₁₄| > 2.22) = 2P(t₁₄ > 2.22) < 0.05.

Question 10

The breaking strengths of a certain brand of marine rope follow a normal distribution, with unknown variance. To test whether the mean breaking strength is 8 units, 28 lengths of the rope were tested. From this sample, the observed value of the test statistic was τ_obs = 2.05. The appropriate t distribution and approximate P-value are:

Your answer is correct.

The P-value is P(|t₂₇| > 2.05) = 2P(t₂₇ > 2.05) ≈ 2 × 0.025.

Not correct. Choice (b) is false.

Try again. Notice that we are conducting a two-sided test.

Not correct. Choice (c) is false.

Try again. The test statistic is not t₂₈.

Not correct. Choice (d) is false.

Try again. The test statistic is not t₈.

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Questions:

right first attempt
right
wrong

a)	H₀ : μ = 5;H₁ : μ > 5	b)	H₀ : μ = 5;H₁ : μ = 5.9
c)	H₀ : x = 5;H₁ : x > 5.9	d)	H₀ : μ = 5;H₁ : μ≠5

a)		The P-value is P(Z ≥ 1.8) = 0.0359 and there is strong evidence that the mean exceeds 5 units.
b)		The P-value is P(Z ≥ 0.45) = 0.3264 which indicates the data are consistent with a mean of 5 units.
c)		The P-value is P(Z ≥ 3.6) = 0.0002 and there is strong evidence that the mean exceeds 5 units.
d)		The P-value is P(\|Z\|≥ 1.8) = 0.0718 which does not constitute strong evidence against a mean of 5 units.

a)	τ ~(0,1) and P-value < 0.05.	b)	τ ~ t₁₆ and P-value < 0.05.
c)	τ ~ t₁₅ and P-value < 0.05.	d)	τ ~ t₁₅ and P-value > 0.05.

a)	$H0 : μ = 60$ ; $H1 : μ > 60$	b)	$H0 : μ = 60$ ; $H1 : μ ⁄= 60$
c)	$H0 : μ = 0$ ; $H1 : μ > 0$	d)	$H0 : μ = 0$ ; $H1 : μ ⁄= 0$

a)	$τ = X ~ N (60,1.22)$	b)	$τ = -X-6√0-~ N (0,1) 1.2∕ 15$
c)	$-- τ = X-1.260-~ N (0,1)$	d)	$-X-μ τ = S∕√n ~ t14$

a)	P-value = 0.011.	b)	P-value = 0.989.
c)	P-value = 0.022.	d)	P-value = 0.10

a)		τ_obs = 60.71, P-value = 0.
b)		$τobs = 1.79$ , P-value =P(\|t₁₄\| > 1.79) > 0.05.
c)		$τ = 2.22 obs$ , P-value = P(t₁₄ > 2.22) < 0.025.
d)		$τobs = 2.22$ , P-value = P(\|t₁₄\| > 2.22) < 0.05.

a)	t₂₇ , P-value ≈ 0.05.	b)	t₂₇ , P-value ≈ 0.025.
c)	t₂₈ , P-value ≈ 0.05.	d)	t₈ , P-value ≈ 0.025.

a)	(0, 0.05)	b)	(0.05, 0.10)
c)	(0.10, 1)	d)	(0.25, 1)

a)	c = 1.771	b)	c = 1.645
c)	c = 0.05	d)	c = 1.782

The University of Sydney - School of Mathematics and Statistics

Quiz 9: t-tables; Z-tests and t-tests for a mean

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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