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Solve question 1-9 Chapter 16 Hypothesis testing for the mean: Assignment Name Q...

Jun 17, 2024

Solve question 1-9

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Solution by Steps

step 1

To find the 90% confidence interval for the mean, we use the formula for the confidence interval:

\bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)

step 2

For a 90% confidence interval,

z_{\alpha/2} = 1.645

step 3

Calculate the margin of error:

1.645 \left( \frac{7.9}{\sqrt{48}} \right) = 1.876

step 4

Compute the confidence interval:

86.5 \pm 1.876 = (84.624, 88.376)

# Part (b)

step 1

For a 95% confidence interval,

z_{\alpha/2} = 1.96

step 2

Calculate the margin of error:

1.96 \left( \frac{7.9}{\sqrt{48}} \right) = 2.234

step 3

Compute the confidence interval:

86.5 \pm 2.234 = (84.266, 88.734)

# Part (c)

step 1

For a 99% confidence interval,

z_{\alpha/2} = 2.576

step 2

Calculate the margin of error:

2.576 \left( \frac{7.9}{\sqrt{48}} \right) = 2.937

step 3

Compute the confidence interval:

86.5 \pm 2.937 = (83.563, 89.437)

Answer

(a) (84.624, 88.376)

(b) (84.266, 88.734)

Key Concept

Confidence Interval Calculation

Explanation

The confidence interval is calculated using the sample mean, standard deviation, and the z-value corresponding to the desired confidence level.

Question 2

step 1

To find the 98% confidence interval for the mean, we use the formula for the confidence interval:

\bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)

step 2

For a 98% confidence interval,

z_{\alpha/2} = 2.33

step 3

Calculate the margin of error:

2.33 \left( \frac{12}{\sqrt{38}} \right) = 4.53

step 4

Compute the confidence interval:

68 \pm 4.53 = (63.47, 72.53)

Answer

(63.47, 72.53)

Key Concept

Confidence Interval Calculation

Explanation

The confidence interval is calculated using the sample mean, standard deviation, and the z-value corresponding to the desired confidence level.

Question 3 # Part (a)

step 1

To find the 95% confidence interval for the mean, we use the formula for the confidence interval:

\bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)

step 2

For a 95% confidence interval,

z_{\alpha/2} = 1.96

step 3

Calculate the margin of error:

1.96 \left( \frac{14}{\sqrt{61}} \right) = 3.51

step 4

Compute the confidence interval:

48 \pm 3.51 = (44.49, 51.51)

# Part (b)

step 1

To find the required sample size, we use the formula:

n = \left( \frac{z_{\alpha/2} \cdot s}{E} \right)^2

step 2

For a 95% confidence interval,

z_{\alpha/2} = 1.96

step 3

Calculate the sample size:

n = \left( \frac{1.96 \cdot 14}{5} \right)^2 = 30.75 \approx 31

Answer

(a) (44.49, 51.51)

(b) 31

Key Concept

Confidence Interval and Sample Size Calculation

Explanation

The confidence interval is calculated using the sample mean, standard deviation, and the z-value corresponding to the desired confidence level. The sample size is calculated to achieve a specified margin of error.

Question 4 # Part (a)

step 1

To find the 99% confidence interval for the mean, we use the formula for the confidence interval:

\bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)

step 2

For a 99% confidence interval,

z_{\alpha/2} = 2.576

step 3

Calculate the margin of error:

2.576 \left( \frac{10}{\sqrt{30}} \right) = 4.71

step 4

Compute the confidence interval:

58 \pm 4.71 = (53.29, 62.71)

# Part (b)

step 1

For a 95% confidence interval,

z_{\alpha/2} = 1.96

step 2

Calculate the margin of error:

1.96 \left( \frac{10}{\sqrt{30}} \right) = 3.58

step 3

Compute the confidence interval:

58 \pm 3.58 = (54.42, 61.58)

# Part (c)

step 1

For a 90% confidence interval,

z_{\alpha/2} = 1.645

step 2

Calculate the margin of error:

1.645 \left( \frac{10}{\sqrt{30}} \right) = 3.00

step 3

Compute the confidence interval:

58 \pm 3.00 = (55.00, 61.00)

Answer

(a) (53.29, 62.71)

(b) (54.42, 61.58)

Key Concept

Confidence Interval Calculation

Explanation

The confidence interval is calculated using the sample mean, standard deviation, and the z-value corresponding to the desired confidence level.

Question 5 # Part (a)

step 1

To find the 99% confidence interval for the mean, we use the formula for the confidence interval:

\bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)

step 2

For a 99% confidence interval,

z_{\alpha/2} = 2.576

step 3

Calculate the margin of error:

2.576 \left( \frac{5.7}{\sqrt{47}} \right) = 2.14

step 4

Compute the confidence interval:

14.6 \pm 2.14 = (12.46, 16.74)

# Part (b)

step 1

To find the required sample size, we use the formula:

n = \left( \frac{z_{\alpha/2} \cdot s}{E} \right)^2

step 2

For a 99% confidence interval,

z_{\alpha/2} = 2.576

step 3

Calculate the sample size:

n = \left( \frac{2.576 \cdot 5.7}{1} \right)^2 = 215.4 \approx 216

Answer

(a) (12.46, 16.74)

(b) 216

Key Concept

Confidence Interval and Sample Size Calculation

Explanation

Question 6

step 1

The null hypothesis is

H_0: \mu = 100

and the alternative hypothesis is

H_1: \mu \neq 100

step 2

The p-value is 0.0419, which is less than the significance level

\alpha = 0.05

step 3

Since the p-value is less than

\alpha

, we reject the null hypothesis

Answer

Reject the null hypothesis

Key Concept

Hypothesis Testing

Explanation

If the p-value is less than the significance level, we reject the null hypothesis.

Question 7 # Part (a)

step 1

The null hypothesis is

H_0: \mu = 12

and the alternative hypothesis is H_1: \mu < 12

step 2

The p-value is 0.035, which is less than the significance level

\alpha = 0.05

step 3

Since the p-value is less than

\alpha

, we reject the null hypothesis

# Part (b)

step 1

For a two-tailed test, the p-value is doubled:

2 \times 0.035 = 0.07

step 2

The p-value for the two-tailed test is 0.07, which is greater than the significance level

\alpha = 0.05

step 3

Since the p-value is greater than

\alpha

, we fail to reject the null hypothesis

Answer

(a) Reject the null hypothesis

(b) Fail to reject the null hypothesis

Key Concept

Hypothesis Testing

Explanation

If the p-value is less than the significance level, we reject the null hypothesis. For a two-tailed test, the p-value is doubled.

Question 8 # Part (a)

step 1

The null hypothesis is

H_0: \mu = 5

and the alternative hypothesis is H_1: \mu > 5

# Part (b)

step 1

The dot plot shows the distribution of sample means under the null hypothesis

step 2

The observed sample mean is 5.2, which is greater than the null hypothesis mean of 5

step 3

The p-value is the proportion of sample means greater than or equal to 5.2

step 4

From the dot plot, the p-value is approximately 0.03

# Part (c)

step 1

The p-value of 0.03 indicates moderate evidence against the null hypothesis

# Part (d)

step 1

Since the p-value is less than the significance level

\alpha = 0.05

, we reject the null hypothesis

Answer

(a)

H_0: \mu = 5

, H_1: \mu > 5

(b) p-value ≈ 0.03

(d) Reject the null hypothesis

Key Concept

Hypothesis Testing and p-value Interpretation

Explanation

The p-value is the probability of observing the sample mean under the null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

Question 9 # Part (a)

step 1

The null hypothesis is

H_0: \mu = 60

and the alternative hypothesis is

H_1: \mu \neq 60

step 2

The p-value is 0.0339, which is less than the significance level

\alpha = 0.05

step 3

Since the p-value is less than

\alpha

, we reject the null hypothesis

step 4

The p-value is 0.0339, which is greater than the significance level

\alpha = 0.01

step 5

Since the p-value is greater than

\alpha

, we fail to reject the null hypothesis

# Part (b)

step 1

If the sample size is reduced to 25, the standard error increases, leading to a higher p-value

step 2

With a higher p-value, the likelihood of rejecting the null hypothesis decreases

Answer

(a) (i) Reject the null hypothesis (ii) Fail to reject the null hypothesis

(b) (i) Higher p-value (ii) Likely to fail to reject the null hypothesis

Key Concept

Hypothesis Testing and Sample Size Effect

Explanation

The p-value is the probability of observing the sample mean under the null hypothesis. A larger sample size results in a smaller standard error and a lower p-value.