Question

The earth's temperature can be measured using either ground-based sensors or infrared-sensing devices mounted in aircraft or space satellites. Ground-based sensoring is very accurate but tedious, while infrared-sensoring appears to introduce a bias into the temperature readings-that is, the average temperature reading may not be equal to the average obtained by ground-based sensoring. To determine the bias, readings were obtained at five different locations using both ground- and air-based temperature sensors. The readings (in degrees Celsius) are listed here. Location Ground Air 47.5 46.7 45.5 48.2 36,4 38.1 31.0 32.8 24.5 26.1 (a) Do the data present sufficient evidence to indicate a bias in the air-based temperature readings? Explain. (use α-0.05. Use μground-Hair-μο State the null and alternative hypotheses Ho: Hoキ0 versus Ha: Ha = 0 0 versus Ha: μd 0 Ho: μd = State the test statistic. (Round your answer to three decimal places.) t1.887 State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.) t > 2.776 -2.776 State the conclusion Ho is rejected. There is insufficient evidence to conclude that there is a bias in the air-based temperature readings. Ho is not rejected. There is insufficient evidence to conclude that there is a bias in the air-based temperature readings. Ho is not rejected. There is sufficient evidence to conclude that there is a bias in the air-based temperature readings. О но is rejected. There is sufficient evidence to conclude that there is a bias in the air-based temperature readings b) Estimate the difference in mean temperatures in degrees Celsius between ground sensors using a 95% confidence interva Round your answers。 and air base џа μ round .ai three decimal places.) 4.25 XC to 0.81 (c) How many paired observations are required to estimate the difference between mean temperatures for ground- versus air-based sensors correct to within 0.2°C, with probability approximately equal to 0.95? (Round your answer up to the nearest whole number.) X paired observations

Answer 1

We conduct T-test for Paired Samples as :

Difference Sample 1 - Sample 2 Sample 1 Sample 2 46.7 47.5 0.8 48.2 -2.7 45.5 38.1 36.4 31 32.8 1.8 24.5 26.1 1.6 36.82 Average 38.54 0.676 St. Dev 9.471 9.506 From the sample data, it is found that the corresponding sample means are X1 36.82 Xo 38.54 Also, the provided sample standard deviations are 81-9.471 82 = 9.506

and the sample size is n = 5, For the score differences we have =-1.72 sp = 0.676 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested This corresponds to a two-tailed test, for which a t-test for two paired samples be used. Based on the information provided, the significance level is α-005, and the degrees of freedom are df = 4. Hence, it is found that the critical value for this two-tailed test is te 2.776, for a 0.05 and df-4 The rejection region for this two-tailed test is R = {t: |t| > 2.776) (3)_Test Statistics The t-statistic is computed as shown in the following formula: t = -5.689 /v/n = 0.676/v/5

(4) Decision about the null hypothesis Since it is observed that tl rejected 5.689 > tcー2.776, it is then concluded that the null hypothesis is Using the P-value approach: The p-value is p 0.0047, and since p 0.00470.05, it is It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μι is different than μ , at the 0.05 significance level The 95% confidence interval is-2.559 < μD <-0.881.

(a)

Correct option : Ho : $\small \mu_d$ = 0 versus Ha : $\small \mu_d$ $\small \neq$ 0

Test statistics = - 5.689

Rejection Region :

t > 2.776

t < - 2.776  

Conclusion :

Correct option : H₀ is rejected.There is sufficient evidence to conclude that there is a bias in the air-based temperature readings.

(b)

95% confidence interval is :

- 2.559 ^oC to - 0.881 ^oC

(c)

We are given :

M.E. = Margin of error = 0.2

= (ME). ( 2.776 * 0.676 )- = 89 rl 0.2

Hence, 89 paired observations