Question

1. The planning fallacy is the tendency that people estimate the time required for the completion of a project to be shorter than it really is. To test whether people in your company exhibit planning fallacy, a sample of 36 workers was selected. The workers were asked how long it would take them to perform a certain task. Their answer was 40 minutes. Then, they were asked to perform the task. The time they needed to complete it had mean 43 minutes. Suppose the standard deviation of completion time (i.e. the population standard deviation) is known to be 5 minutes:

1. Is there sufficient evidence that the population mean time to complete the task is greater than 40 minutes? Use a level of significance of 0.05. Do these results support the claim, that people in your company exhibit a planning fallacy?

2. Is there sufficient evidence that the population mean time to complete the task is different from 40 minutes? Use a level of significance of 0.01

3. Repeat Section a, but assume that the standard deviation of the time required to complete the task is known to be 9 minutes instead of 5 minutes. In addition, perform the test using a significance level of 0.01. Please also report the p-value.

Please submit complete solutions for all sections. You can use a calculator and the normal distribution table.

Answer 1

We will be using z-test for population mean since we have the population standard deviation.

Listing down the stats

$n=36\;\bar{x}=43\;\sigma=5\;\mu_{0}=40$ (null mean)

All the critical values and the p-values are found using normal distribution, percentage tables and excel functions 'normsinv' and 'normsdist'

(a) We have to test whether the population mean time to complete the task is greater than 40 minutes

$H_{0}:\mu\leq \mu_{0}$ (The population mean time to complete the task is less than 40 minutes that is people in your company do not exhibit a planning fallacy)

VS

$H_{1}:\mu> \mu_{0}$ (The population mean time to complete the task is greater than 40 minutes that is people in your company exhibit a planning fallacy))

Test statistic: $\frac{\bar{x}-\mu_{0}}{\sigma/\sqrt{n}}$

= $\frac{43-40}{5/\sqrt{36}}$

= $3.6$

We have to test at $\alpha$ = 0.05

Critical value at 0.05 = $Z_{\alpha}$ (We are only testing in one direction right tailed and hence it is 1-tailed test)

= $Z_{0.05}$

=$1.6449$

Since Test stat > Critical value

Decision: We reject the null hypothesis at 0.05 level of significance and conclude that the population mean time to complete the task is significantly greater than 40 minutes that is people in your company exhibit a planning fallacy.

(b) We have to test whether the population mean time to complete the task is different from 40 minutes

$H_{0}:\mu= \mu_{0}$ (The population mean time to complete the task is 40 minutes)

VS

$H_{1}:\mu\neq \mu_{0}$ (The population mean time to complete the task is not 40 minutes)

Test statistic: $\frac{\bar{x}-\mu_{0}}{\sigma/\sqrt{n}}$

= $\frac{43-40}{5/\sqrt{36}}$

= $3.6$

We have to test at $\alpha$ = 0.01

Critical value at 0.01 = $Z_{\alpha/2}$ (We are testing in both directions (difference) and hence it is 2- tailed test)

= $Z_{0.01/2}=Z_{0.005}$

=$2.5758$

Since Test stat > Critical value

Decision: We reject the null hypothesis at 0.01 level of significance and conclude that the population mean time to complete the task is significantly different from 40 minutes.

(c) We have to test whether the population mean time to complete the task is greater than 40 minutes

but we have population sd = 9 and level of significance = 0.01

$H_{0}:\mu\leq \mu_{0}$ (The population mean time to complete the task is less than 40 minutes that is people in your company do not exhibit a planning fallacy)

VS

$H_{1}:\mu> \mu_{0}$ (The population mean time to complete the task is greater than 40 minutes that is people in your company exhibit a planning fallacy))

Test statistic: $\frac{\bar{x}-\mu_{0}}{\sigma/\sqrt{n}}$

= $\frac{43-40}{9/\sqrt{36}}$

= $2$

We have to test at $\alpha$ = 0.01

Critical value at 0.05 = $Z_{\alpha}$ (We are only testing in one direction right tailed and hence it is 1-tailed test)

= $Z_{0.01}$

=$2.3264$

Since Test stat < Critical value

Checking by way of p -value

p - value = P ( Z > Test Stat)

= P ( Z > 2)

= 1 - P(Z < 2)

= 1 - 0.97725

= 0.02275

Also p value > level of significance (0.01)

p-value is the probability of null hypothesis being true

Decision: We do not reject the null hypothesis at 0.01 level of significance and conclude that the population mean time to complete the task is significantly less than 40 minutes that is people in your company do not exhibit a planning fallacy.

Explanation: The reason we do not reject the null hypothesis is because the variability reduces the test statistic which makes the null hypothesis more likely and also the level of significance is reduced which makes the test stricter and in favour of the null hypothesis.