Packages of sugar bags for Sweeter Sugar Inc. have an average weight of 16 ounces and a standard deviation of 0.3 ounces. The we
1 answer:
Answer:
40.13% probability that 9 randomly selected packages will have an average weight in excess of 16.025 ounces
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random normally distributed variable X, with mean and standard deviation
, the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
What is the probability that 9 randomly selected packages will have an average weight in excess of 16.025 ounces
This is 1 subtracted by the pvalue of Z when X = 16.025. So
By the Central Limit Theorem
has a pvalue of 0.5987
1 - 0.5987 = 0.4013
40.13% probability that 9 randomly selected packages will have an average weight in excess of 16.025 ounces
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Answer:
The p-value of the test is of 0.0287 < 0.05(standard significance level), which means that it can be concluded that less than half of HIV-positive smokers have used a nicotine patch.
Step-by-step explanation:
Test if less than half of HIV-positive smokers have used a nicotine patch:
At the null hypothesis, we test if the proportion is of at least half, that is:
At the alternative hypothesis, we test if the proportion is below 0.5, that is:
The test statistic is:
In which X is the sample mean, is the value tested at the null hypothesis,
is the standard deviation and n is the size of the sample.
0.5 is tested at the null hypothesis:
This means that
In a survey of 444 HIV-positive smokers, 202 reported that they had used a nicotine patch to try to quit smoking.
This means that
Value of the test statistic:
P-value of the test and decision:
The p-value of the test is the probability of finding a sample proportion below 0.455, which is the p-value of z = -1.9.
Looking at the z-table, z = -1.9 has a p-value of 0.0287.
The p-value is of 0.0287 < 0.05(standard significance level), which means that it can be concluded that less than half of HIV-positive smokers have used a nicotine patch.