Illustrate an image of an abstract city skyline with buildings of diverse heights, representing the varying weights of the population. Include an overlay graph showing a normal distribution curve, signifying the population's weight distribution. Indicate a specific point on the curve to represent the mean weight of the sampled group from one of the city's neighborhoods. The image should be devoid of any text or numbers.

The weight of adults in a certain city has a mean of 130 lbs, with standard deviation of

41 lbs. A sample of 25 adults living in one of the kebeles of the city is found to have a
mean weight of 129 lbs. Does this mean the residents of the kebele weigh
significantly less on the basis of the1% level of significance? (Assume normal
population.)
a) State the null and the alternative hypothesis.
b) Based on the information provided would you employ the Z or the t as the
distribution of the test statistic?
c) Give the critical region
d) What is your decision?

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The weight of adults in a certain city has a mean of 130 lbs, with standard deviation of 41 lbs. A sample of 25 adults living in one of the kebeles of the city is found to have a mean weight of 129 lbs. Does this mean the residents of the kebele weigh significantly less on the basis of the1% level of significance? (Assume normal population.)

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125 Stati

Required answer

a) The null hypothesis (H0) is that the residents of the kebele weigh the same as the average weight of adults in the city, while the alternative hypothesis (Ha) is that the residents of the kebele weigh significantly less.

H0: The mean weight of residents in the kebele = 130 lbs
Ha: The mean weight of residents in the kebele < 130 lbs

b) We would use the t-distribution as the sample size (25) is small (<30) and we do not know the population standard deviation.

c) The critical region is the region of extreme sample values which, if the test statistic falls within it, would lead to the rejection of the null hypothesis. In this case, the critical region would be on the left side of the distribution due to the alternative hypothesis being "less than."

Since we are testing at the 1% level of significance, the critical region is the lower 1% of the t-distribution.

d) To make a decision, we need to calculate the t-test statistic using the given information and compare it to the critical value. If the calculated t-test statistic falls within the critical region, we reject the null hypothesis. If it does not fall within the critical region, we fail to reject the null hypothesis.

To calculate the t-test statistic, we use the formula:

t = (sample mean - population mean) / (sample standard deviation / √sample size)

t = (129 - 130) / (41 / √25)
t = -1 / (41 / 5)
t = -5/41

After obtaining the t-test statistic, we compare it to the critical value from the t-distribution table with a degree of freedom (df) equal to sample size - 1 (25 - 1 = 24) at the 1% level of significance.

If the t-test statistic is less than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Since the t-test statistic of -5/41 falls within the critical region (the lower 1% of the t-distribution), we reject the null hypothesis. This means that the residents of the kebele weigh significantly less than the average weight of adults in the city at a 1% level of significance.