Question

9. A soft drink manufacturer claims its sodas contain 30 calories. A random sample of 61 cans contained a mean of 32 calories, with a std. deviation of 3 calories. Test whether the mean is greater than 30 calories (and comment.

I need help with some homework problems I got stuck on. May you please show step by step so I may understand it please. Thank you.

Answer 1

Solution:

Given:

Sample size = n = 61

Sample mean = calories

T= 32

Sample standard deviation = s = 3 calories

We have to test whether the mean is greater than 30 calories or not.

Step 1) State H0 and H1:

Ho :μ = 30

Vs

H, :μ> 30

Step 2) Test statistic:

Since population standard deviation is unknown, we use t test for one population mean.

- s/n

32 – 30 3/61

2 ( 37.810250

0.3841106

t = 5,207

Step 3) t critical value:

df = n -1 = 61 - 1= 60

One tail area ( Since H1 is > type) = Level of significance = 0.05

t Table cum. prob one-tail two-tails 0.50 1.00 0.25 0.50 0.20 0.40 0.15 0.30 0.10 0.20 0.05 0.10 0.025 0.05 0.01 0.02 GANDOO VOORN 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 - 100 1.000 1.376 0.816 1.061 0.785 0.978 0.741 0.941 0.727 0.920 0.7180.906 0.711 0.896 0.706 0.889 0.703 0.883 0.700 0,879 0.697 0.876 0.695 0.873 0.894 0.870 0.692 0.868 0.691 0.866 0.690 0.865 0.689 0.863 0.688 0.862 0.688 0.861 0.687 0.860 0.686 0.85€ 0.686 0.858 0.685 0.858 0.685 0.857 0.684 0.85€ 0.684 0.85€ 0.684 0.855 0.683 0.855 0.683 0.854 0.683 0.854 0.681 0.851 .0 0.548 1.963 1.386 1.250 1.190 1.158 1.134 1.119 1.108 1.100 1.093 1.088 1.083 1.079 1.076 1.074 1.071 1.069 1.067 1.066 1.064 1.063 1.061 1.060 1.050 1.058 1.058 4067 1.057 1.066 1065 1.055 1.055 1.050 20 3.078 1.886 1.638 1.533 1.478 1.440 1.415 1.397 1.383 1.372 1.383 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1311 1.310 1.303 1 2 6.14 2.20 2353 2.32 2515 1943 1 595 1560 1533 1 312 17796 1782 1 771 1781 1753 1 746 1740 12.71 31.82 4.303 0.005 3.182 4.541 2.776 3.747 2571 3.365 2.447 3.143 2.365 2.098 2.308 2.896 2.262 2821 2.229 2.784 2.2012.718 2.179 2.681 2.160 2.850 2.145 2.024 2.131 2.802 2.120 2.583 2.110 22567 2.101 2.552 2.093 2.530 2.086 2.528 2.080 2.518 2.074 2.508 2.069 2.500 2.064 2402 2.060 2485 2.056 2479 2.052 2.473 2467 2.045 2.462 2.042 2.457 2.021 2.423 2.000 2.390 1734 1729 1.725 .721 .717 .714 .711 .708 .708 .703 .701 1.099 1697 1684 1.671 60

t critical value = 1.671

Step 4) Decision Rule:

Reject null hypothesis H0, if t test statistic value > t critical value =1.671, otherwise we fail to reject H0.

Since  t test statistic value = > t critical value =1.671, we reject null hypothesis H0.

t = 5,207

Step 5) Conclusion:

At 0.05 level of significance we have sufficient evidence to conclude that the mean calories. of sodas is greater than 30 calories.