Please answer this question
1 answer:
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Answer:
a) 0.047
b) 50% probability that the sample proportion of smart phone users is greater than 0.33.
c) 33.39% probability that the sample proportion is between 0.19 and 0.31
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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 normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation
In this question, we have that:
a) What would the standard deviation of the sampling distribution of the proportion of the smart phone users be?
b) What is the probability that the sample proportion of smart phone users is greater than 0.33?
This is 1 subtracted by the pvalue of Z when X = 0.33. So
By the Central Limit Theorem
has a pvalue of 0.5
1 - 0.5 = 0.5
50% probability that the sample proportion of smart phone users is greater than 0.33.
c) What is the probability that the sample proportion is between 0.19 and 0.31?
This is the pvalue of Z when X = 0.31 subtracted by the pvalue of Z when X = 0.19. So
X = 0.31
has a pvalue of 0.3354
X = 0.19
has a pvalue of 0.0015
0.3354 - 0.0015 = 0.3339
33.39% probability that the sample proportion is between 0.19 and 0.31