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The manager of a donut store believes that 35% of the customers are first-time customers. A random sample of 150 customers will be used to e
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The manager of a donut store believes that 35% of the customers are first-time customers. A random sample of 150 customers will be used to estimate the proportion of first-time customers. Assuming this belief is correct, what is the probability that the sample proportion will be between 0.2 and 0.4
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Mathematics
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2021-08-06T04:25:29+00:00
2021-08-06T04:25:29+00:00 1 Answers
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Answer:
0.8996 = 89.96% probability that the sample proportion will be between 0.2 and 0.4
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes 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 ![Rendered by QuickLaTeX.com s = \sqrt{\frac{p(1-p)}{n}}](https://documen.tv/wp-content/ql-cache/quicklatex.com-00023124723f30bde741bd4eedc6930c_l3.png)
The manager of a donut store believes that 35% of the customers are first-time customers.
This means that![Rendered by QuickLaTeX.com p = 0.35](https://documen.tv/wp-content/ql-cache/quicklatex.com-326f7952961f5356f085235866a8f9f0_l3.png)
Sample of 150 customers
This means that![Rendered by QuickLaTeX.com n = 150](https://documen.tv/wp-content/ql-cache/quicklatex.com-a651d6d73e09cbd18174a952932c80e6_l3.png)
Mean and standard deviation:
What is the probability that the sample proportion will be between 0.2 and 0.4?
p-value of Z when X = 0.4 subtracted by the p-value of Z when X = 0.2.
X = 0.4
By the Central Limit Theorem
X = 0.2
0.8997 – 0.0001 = 0.8996
0.8996 = 89.96% probability that the sample proportion will be between 0.2 and 0.4