I have tried to find the p value and z statitcs for each of the firms below and I am so confused becuase my answers arent making any sense. The randome sample is 40, the hypothesis we are testing is a lower tail test 35, pop st deviation is 6 and level of sig is 0.03. I have tried to find the p values but it doesnt make sense because my answers look wrong because for firm 1 the p value i got was 0.99 and the z stat. was 2.65 and for firm 2 the p value was 0.04 and the z st was -1.73. I kniw this is wrong because the values for the z score should all be negative because its a lower tail test. Please help - below is the data for the firms from excel, I honeslty need help and I want to understand how to do it
firm1 firm2
31.92 35.35
43.27 29.74
43.36 43.17
49.18 40.67
40.21 28.69
29.35 35.17
52.18 30.29
33.45 27.74
32.89 36.61
34.88 27.21
42.18 33.08
37.91 35.46
36.69 39.80
31.41 26.84
37.02 23.60
30.74 32.20
33.07 29.36
48.78 27.42
42.34 32.98
40.53 36.51
35.44 34.42
37.34 40.33
31.32 28.53
44.47 48.18
38.03 32.89
32.82 34.65
21.66 27.57
39.92 36.53
36.10 32.75
37.79 36.92
40.01 38.27
45.74 31.94
48.74 37.61
37.07 34.39
19.80 33.73
30.52 34.99
37.45 32.82
42.59 25.97
37.20 32.81
35.47 26.83
To find the p-value and z-statistic for each firm, you need to perform a hypothesis test. Given that the hypothesis is a lower tail test with a significance level (α) of 0.03, the steps to calculate these values are as follows:
1. Calculate the sample mean (x̄) for each firm.
2. Calculate the sample standard deviation (s) for each firm.
3. Calculate the standard error of the mean (SE) for each firm, which is given by the formula SE = s/√n, where n is the sample size.
4. Calculate the test statistic (z) for each firm, which is given by the formula z = (x̄ - μ) / SE, where μ is the hypothesized population mean (35 in this case).
5. Look up the critical value (z-critical) for a lower tail test at the given significance level (α) using a standard normal distribution table or a calculator. In this case, with α = 0.03, the z-critical value can be found to be approximately -1.88.
6. Compare the test statistic (z) to the critical value (z-critical) to determine whether to reject or fail to reject the null hypothesis.
- If z ≤ z-critical, reject the null hypothesis.
- If z > z-critical, fail to reject the null hypothesis.
7. Calculate the p-value for each firm, which represents the probability of obtaining a test statistic as extreme as the one observed under the null hypothesis.
- For a lower tail test, the p-value is the area to the left of the observed test statistic (z) on the standard normal distribution.
- P-values can be obtained using a standard normal distribution table or a calculator.
Now, let's calculate the p-value and z-statistic for each firm using the given data:
Firm 1:
- Sample mean (x̄1): Calculate the mean of the first firm's data.
- Sample standard deviation (s1): Calculate the standard deviation of the first firm's data.
- Standard error of the mean (SE1): Calculate the standard error of the first firm's data using the formula (SE = s1/√n).
- Z-statistic (z1): Calculate the z-statistic for the first firm using the formula (z1 = (x̄1 - μ)/SE1).
- P-value (p1): Calculate the p-value for the first firm using the standard normal distribution table or a calculator.
Repeat the above steps for Firm 2 using the corresponding data.
By following these steps, you can obtain the correct p-value and z-statistic for each firm and make sense of the results.