It has been reported that 48% of teenagers play video games on their phones. A random sample of 60 teenages is drawn. Find the probability that the proportion of teenagers in the sample who play videos games on their phone is less than 0.499 .
Write only a number as your answer. Round to 4 decimal places (for example 0.3748). Do not write as a percentage.
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To find the probability that the proportion of teenagers in the sample who play video games on their phone is less than 0.499, we need to use the sampling distribution of proportions.
Given that 48% (or 0.48) of teenagers play video games on their phones, we can assume that the population proportion is 0.48.
The sample size is 60, so we have n = 60.
To calculate the probability, we can use the sampling distribution of proportions. The formula for the standard deviation of the sampling distribution is:
Standard Deviation = sqrt((p * (1 - p)) / n)
where p is the population proportion and n is the sample size.
The standard deviation for this sampling distribution is:
Standard Deviation = sqrt((0.48 * (1 - 0.48)) / 60)
Now, we can standardize the proportion we are interested in using the standard deviation:
Z = (sample proportion - population proportion) / sqrt((p * (1 - p)) / n)
Substituting the values:
Z = (0.499 - 0.48) / sqrt((0.48 * (1 - 0.48)) / 60)
Calculating this expression will give us the z-score. Then, we can use a Z-table or a calculator to find the corresponding probability.
Once you have the z-score, you can look up the probability corresponding to that z-score in a standard normal distribution table or use a calculator with a normal distribution function to find the probability.
Finally, round your answer to 4 decimal places before writing it down.