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An investment website can tell what devices are used to access their site. The site managers wonder whether they should enhance the facilities for trading via smartphones so they want to estimate the proportion of users who access the site that way. They draw a random sample of 131 investors from their customers. Suppose that the true proportion of smartphone users is 26 %. Complete parts​ a) through​ c) below.

a. What would the managers expect the shape of the sampling distribution for the sample proportion to​ be?
​b. What would be the mean of this sampling​ distribution?
​c. What would be the standard deviation of the sampling​distribution?

Respuesta :

Answer:

a) Unimodal and symmetric (Normal distribution)

b) Mean of the sampling distribution = 0.26

c) Standard deviation of the sampling distribution = 0.0383

Step-by-step explanation:

Sample size, n = 131

Proportion of smart phone users = 26%

p = 26/100 = 0.26

q = 1 - p = 0.74

np = 131 * 0.26 = 34.06

nq = 131 * 0.74 = 96.94

Both the sizes of those that use smart phone and those that do not use are large enough to be modeled by normal distribution.

The shape is therefore unimodal and symmetric

b) What would be the mean of this sampling distribution?

Mean, μ = the proportion of those that use smart phones

μ = 0.26

c) Standard deviation of the sampling distribution

[tex]\sigma = \sqrt{\frac{pq}{n} } \\\sigma = \sqrt{\frac{(0.26*0.74)}{131} } \\\sigma = 0.0383[/tex]

Chimara

Answer:

(A) The managers would expect the shape of the sampling distribution of the sampling proportion to be normal.

(B) The mean of this sampling distribution would be 34.06 investors

(C) The standard deviation of the sampling distribution would be 0.038 (to 3 decimal places)

Step-by-step explanation:

(A) The sample is a large random sample, that is, sample size is greater than 30. Sample size is represented by N so in this case, N=131

According to the Central Limit Theorem, the shape of the sampling distribution of the mean sample proportion approaches a normal distribution as the sample size N increases. This applies here as the sample size is very large.

(B) The mean of the sampling distribution of the proportion of smartphone users is EQUAL TO the population mean of the proportion of smartphone users. Hence, the mean = 26% of 131 which is = 34.06 investors.

(C) The standard deviation of the sampling distribution of the mean of smartphone users is gotten from the probability of smartphone users and the probability of other-device users.

The probability that a randomly selected investor uses a smartphone to access the website is 0.26 (26%).

The probability that a randomly selected investor uses other devices to access the website is (1-0.26) which is = 0.74

The standard deviation of the sampling distribution would then be the square root of the variance.

√[(0.26×0.74) ÷ 131]

= √0.1924/131

= √0.00146

= 0.038 (approximated to 3 decimal places)

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