Question

1. According to a government spokesman, the salary of a nurse has a mean of £23,000. You believe this information is incorrect; so you take a sample of 54 nurses and find that the sample mean salary is £22,600 with a standard deviation of £1600. a) Test the government’s claim at the 5% level of significance. b) A union spokesman now claims that nurses are paid less than £23,000. Test this claim. c) Are these 2 claims the same? Explain.

2. The mean satisfaction score of employees in a firm is 947 with a standard deviation of 205. If you select 60 employees randomly, what is the probability that the mean is below 897?

3. You have data for a sample of 100 exams taken by 16 year old students. You are told that the grades follow a normal distribution, with a variance of 25. You want to use this data to test the null hypothesis that μ=10, vs Ha μ>10.

1. What is the power of the test if the true population mean is μ=10.5 and you want a 5% significance level?
2. What is the power of the test if the true population mean is μ=11 and you want a 5% significance level?
3. what would the sample size need to be to be achieve a power of 80% when testing that μ>10 and μ=10.5 and you want to have a probability of Type I error of 5%

4. You want to test whether the quarter of birth affects grades at university. Assuming that births are randomly distributed throughout the year (this might not be correct in real life) you collect data on a sample of 100 students and get the following:

Quarter birth	Q1	Q2	Q3	Q4
Average grade	65	62	63	58
N	25	27	28	20

Compute the appropriate test using a 5% significance level

Hint: Don’t forget to reweight the test of each category by the number of observations, as each cell represents the mean Xi for m observations.

5. You took a sample of 5 phones and tested their battery life. You got the following battery life (in months) 24, 35, 22, 27, 20.

1. You want to test whether the variance in battery life is greater than 30 months squared?

2. You conduct a new survey of phones, and among the 7 phones you survey this time, you report the following battery life. 36, 18, 25, 22, 31, 28, 40

You want to test whether the variance in this sample is greater than in your first survey.

Use http://www.socr.ucla.edu/applets.dir/f_table.html Note this provides only the right tail, so make sure to build your F-Test accordingly with the largest value on the nominator.

Answer 1

Solution:

1(a) The null and alternative hypotheses are as follows:

H_{​​​​​​0} : μ = £23,000 i.e. The population mean salary of nurses is £23,000.

H_{​​​​​1} : μ ≠ £23,000 i.e. The population mean salary of nurses is not equal to £23,000.

To test hypothesis we shall use one sample t-test. The test statistic is given as follows:

Where, x̄ is sample mean, μ is hypothesized value of population mean, s is sample standard deviation and n is sample size.

We have, x̄ = £22600, μ = £23000, s = £1600 and n = 54

The value of the test statistic is -1.8371

Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic. The two-tailed p-value for the test statistic is given as follows:

p-value = 2.P(T > |t|)

We have, |t| = 1.8371

Hence, p-value = 2.P(Z > 1.8371)

p-value = 0.0718

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​0}) at given significance level

We have, p-value = 0.0718 and significance level = 5% = 0.05

(0.0718 > 0.05)

Since, p-value is greater than the significance level of 5%, therefore we shall be fail to reject the null hypothesis (H_{​​​​​​0}) at significance level of 5%.

Conclusion: At 5% significance level, there is enough evidence to support the government's claim that the salary of a nurse has a mean of £23,000.

b)

The null and alternative hypotheses are as follows:

H_{​​​​​​0} : μ = £23,000 i.e. The population mean salary of nurses is £23,000.

H_{​​​​​1} : μ < £23,000 i.e. The population mean salary of nurses is less than £23,000.

To test hypothesis we shall use one sample t-test. The test statistic is given as follows:

Where, x̄ is sample mean, μ is hypothesized value of population mean, s is sample standard deviation and n is sample size.

We have, x̄ = £22600, μ = £23000, s = £1600 and n = 54

The value of the test statistic is -1.8371

Since, our test is left-tailed test, therefore we shall obtain left-tailed p-value for the test statistic. The left-tailed p-value for the test statistic is given as follows:

p-value = P(T < t)

p-value = P(Z < -1.8371)

p-value = 0.0359

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​0}) at given significance level

We have, p-value = 0.0359 and significance level = 5% = 0.05

(0.0359 < 0.05)

Since, p-value is less than the significance level of 5%, therefore we shall reject the null hypothesis (H_{​​​​​​0}) at significance level of 5%.

Conclusion: At 5% significance level, there is enough evidence to support the union spokesman's claim that nurses are paid less than £23,000.

c) These claims are not same. One claim said that nurses salary has mean of £23,000 and second claim said that mean salary of nurses is less than £23,000. When we test the government's claim, then our test is two-tailed test and when we test the union spokesman's claim, then our test is left-tailed test.